| L(s) = 1 | − 5·3-s + 11·5-s + 8·9-s + 11-s + 7·13-s − 55·15-s + 26·17-s + 3·19-s + 9·23-s + 53·25-s + 4·27-s − 2·29-s + 2·31-s − 5·33-s − 2·37-s − 35·39-s + 28·41-s − 5·43-s + 88·45-s − 18·47-s − 130·51-s − 53-s + 11·55-s − 15·57-s − 5·59-s − 17·61-s + 77·65-s + ⋯ |
| L(s) = 1 | − 2.88·3-s + 4.91·5-s + 8/3·9-s + 0.301·11-s + 1.94·13-s − 14.2·15-s + 6.30·17-s + 0.688·19-s + 1.87·23-s + 53/5·25-s + 0.769·27-s − 0.371·29-s + 0.359·31-s − 0.870·33-s − 0.328·37-s − 5.60·39-s + 4.37·41-s − 0.762·43-s + 13.1·45-s − 2.62·47-s − 18.2·51-s − 0.137·53-s + 1.48·55-s − 1.98·57-s − 0.650·59-s − 2.17·61-s + 9.55·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 7^{18}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 7^{18}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(35.18842960\) |
| \(L(\frac12)\) |
\(\approx\) |
\(35.18842960\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 5 T + 17 T^{2} + 41 T^{3} + 95 T^{4} + 193 T^{5} + 367 T^{6} + 193 p T^{7} + 95 p^{2} T^{8} + 41 p^{3} T^{9} + 17 p^{4} T^{10} + 5 p^{5} T^{11} + p^{6} T^{12} \) |
| 5 | \( 1 - 11 T + 68 T^{2} - 59 p T^{3} + 1009 T^{4} - 2854 T^{5} + 6891 T^{6} - 2854 p T^{7} + 1009 p^{2} T^{8} - 59 p^{4} T^{9} + 68 p^{4} T^{10} - 11 p^{5} T^{11} + p^{6} T^{12} \) |
| 11 | \( 1 - T + 40 T^{2} - 45 T^{3} + 830 T^{4} - 923 T^{5} + 11003 T^{6} - 923 p T^{7} + 830 p^{2} T^{8} - 45 p^{3} T^{9} + 40 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \) |
| 13 | \( 1 - 7 T + 64 T^{2} - 301 T^{3} + 1660 T^{4} - 6167 T^{5} + 26265 T^{6} - 6167 p T^{7} + 1660 p^{2} T^{8} - 301 p^{3} T^{9} + 64 p^{4} T^{10} - 7 p^{5} T^{11} + p^{6} T^{12} \) |
| 17 | \( 1 - 26 T + 22 p T^{2} - 3667 T^{3} + 27017 T^{4} - 155523 T^{5} + 715711 T^{6} - 155523 p T^{7} + 27017 p^{2} T^{8} - 3667 p^{3} T^{9} + 22 p^{5} T^{10} - 26 p^{5} T^{11} + p^{6} T^{12} \) |
| 19 | \( 1 - 3 T + 40 T^{2} - 83 T^{3} + 1135 T^{4} - 2550 T^{5} + 28311 T^{6} - 2550 p T^{7} + 1135 p^{2} T^{8} - 83 p^{3} T^{9} + 40 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 23 | \( 1 - 9 T + 132 T^{2} - 871 T^{3} + 7316 T^{4} - 36641 T^{5} + 221101 T^{6} - 36641 p T^{7} + 7316 p^{2} T^{8} - 871 p^{3} T^{9} + 132 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \) |
| 29 | \( 1 + 2 T + 80 T^{2} + 60 T^{3} + 3771 T^{4} + 2561 T^{5} + 131159 T^{6} + 2561 p T^{7} + 3771 p^{2} T^{8} + 60 p^{3} T^{9} + 80 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \) |
| 31 | \( 1 - 2 T + 108 T^{2} - 92 T^{3} + 6283 T^{4} - 4287 T^{5} + 241895 T^{6} - 4287 p T^{7} + 6283 p^{2} T^{8} - 92 p^{3} T^{9} + 108 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \) |
| 37 | \( 1 + 2 T + 104 T^{2} + 12 p T^{3} + 4754 T^{4} + 36718 T^{5} + 167257 T^{6} + 36718 p T^{7} + 4754 p^{2} T^{8} + 12 p^{4} T^{9} + 104 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \) |
| 41 | \( 1 - 28 T + 498 T^{2} - 6335 T^{3} + 64485 T^{4} - 537691 T^{5} + 3761283 T^{6} - 537691 p T^{7} + 64485 p^{2} T^{8} - 6335 p^{3} T^{9} + 498 p^{4} T^{10} - 28 p^{5} T^{11} + p^{6} T^{12} \) |
| 43 | \( 1 + 5 T + 234 T^{2} + 997 T^{3} + 23728 T^{4} + 82489 T^{5} + 1334207 T^{6} + 82489 p T^{7} + 23728 p^{2} T^{8} + 997 p^{3} T^{9} + 234 p^{4} T^{10} + 5 p^{5} T^{11} + p^{6} T^{12} \) |
| 47 | \( 1 + 18 T + 369 T^{2} + 4068 T^{3} + 47104 T^{4} + 368849 T^{5} + 3001113 T^{6} + 368849 p T^{7} + 47104 p^{2} T^{8} + 4068 p^{3} T^{9} + 369 p^{4} T^{10} + 18 p^{5} T^{11} + p^{6} T^{12} \) |
| 53 | \( 1 + T + 187 T^{2} - 25 T^{3} + 15082 T^{4} - 18348 T^{5} + 845921 T^{6} - 18348 p T^{7} + 15082 p^{2} T^{8} - 25 p^{3} T^{9} + 187 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \) |
| 59 | \( 1 + 5 T + 234 T^{2} + 552 T^{3} + 23188 T^{4} + 16167 T^{5} + 1522587 T^{6} + 16167 p T^{7} + 23188 p^{2} T^{8} + 552 p^{3} T^{9} + 234 p^{4} T^{10} + 5 p^{5} T^{11} + p^{6} T^{12} \) |
| 61 | \( 1 + 17 T + 362 T^{2} + 4395 T^{3} + 54846 T^{4} + 491969 T^{5} + 4449063 T^{6} + 491969 p T^{7} + 54846 p^{2} T^{8} + 4395 p^{3} T^{9} + 362 p^{4} T^{10} + 17 p^{5} T^{11} + p^{6} T^{12} \) |
| 67 | \( 1 - 22 T + 418 T^{2} - 5190 T^{3} + 60848 T^{4} - 555512 T^{5} + 5004215 T^{6} - 555512 p T^{7} + 60848 p^{2} T^{8} - 5190 p^{3} T^{9} + 418 p^{4} T^{10} - 22 p^{5} T^{11} + p^{6} T^{12} \) |
| 71 | \( 1 - 2 T + 234 T^{2} - 676 T^{3} + 28488 T^{4} - 85952 T^{5} + 2400741 T^{6} - 85952 p T^{7} + 28488 p^{2} T^{8} - 676 p^{3} T^{9} + 234 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \) |
| 73 | \( 1 - 12 T + 325 T^{2} - 2253 T^{3} + 37839 T^{4} - 163328 T^{5} + 2844215 T^{6} - 163328 p T^{7} + 37839 p^{2} T^{8} - 2253 p^{3} T^{9} + 325 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \) |
| 79 | \( 1 - 2 T + 279 T^{2} - 857 T^{3} + 39985 T^{4} - 127480 T^{5} + 3796211 T^{6} - 127480 p T^{7} + 39985 p^{2} T^{8} - 857 p^{3} T^{9} + 279 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \) |
| 83 | \( 1 + 7 T + 190 T^{2} + 1540 T^{3} + 28127 T^{4} + 190708 T^{5} + 2762999 T^{6} + 190708 p T^{7} + 28127 p^{2} T^{8} + 1540 p^{3} T^{9} + 190 p^{4} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12} \) |
| 89 | \( 1 - 39 T + 1048 T^{2} - 20056 T^{3} + 307602 T^{4} - 3813689 T^{5} + 39558571 T^{6} - 3813689 p T^{7} + 307602 p^{2} T^{8} - 20056 p^{3} T^{9} + 1048 p^{4} T^{10} - 39 p^{5} T^{11} + p^{6} T^{12} \) |
| 97 | \( 1 - 35 T + 925 T^{2} - 16583 T^{3} + 255697 T^{4} - 3147865 T^{5} + 34242405 T^{6} - 3147865 p T^{7} + 255697 p^{2} T^{8} - 16583 p^{3} T^{9} + 925 p^{4} T^{10} - 35 p^{5} T^{11} + p^{6} T^{12} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.25385831083475012574044426033, −3.85546082467731594335812629081, −3.78324209280695280589567857334, −3.68772135725835318909120186740, −3.64907812623951497495987693690, −3.53177431098342580443447768762, −3.38645152761329678884106856759, −3.01758235590820966957881802656, −2.99107031907148442558379417765, −2.81475531758544585768994378175, −2.76104721432646765974355199179, −2.70484366951516735666753635541, −2.53471069855902141897648455913, −2.04798562296007547384675573024, −1.98560863609180196661912997591, −1.85285926603468784808157195322, −1.62116661978928907742423152626, −1.61532422446072915473834617514, −1.34379348336759652730420111946, −1.29270415756848486826598616582, −1.03588190125919868110574318827, −0.891770440431529069576219877921, −0.792997018017851689006518677379, −0.63083043584187850715730564522, −0.41625987920696352091302090990,
0.41625987920696352091302090990, 0.63083043584187850715730564522, 0.792997018017851689006518677379, 0.891770440431529069576219877921, 1.03588190125919868110574318827, 1.29270415756848486826598616582, 1.34379348336759652730420111946, 1.61532422446072915473834617514, 1.62116661978928907742423152626, 1.85285926603468784808157195322, 1.98560863609180196661912997591, 2.04798562296007547384675573024, 2.53471069855902141897648455913, 2.70484366951516735666753635541, 2.76104721432646765974355199179, 2.81475531758544585768994378175, 2.99107031907148442558379417765, 3.01758235590820966957881802656, 3.38645152761329678884106856759, 3.53177431098342580443447768762, 3.64907812623951497495987693690, 3.68772135725835318909120186740, 3.78324209280695280589567857334, 3.85546082467731594335812629081, 4.25385831083475012574044426033
Plot not available for L-functions of degree greater than 10.
