| L(s) = 1 | − 3·3-s − 3·7-s + 8-s + 3·11-s − 3·13-s − 3·17-s − 9·19-s + 9·21-s − 3·23-s − 3·24-s + 12·27-s − 18·29-s + 3·31-s − 9·33-s + 12·37-s + 9·39-s − 21·41-s − 18·43-s + 9·47-s + 21·49-s + 9·51-s − 9·53-s − 3·56-s + 27·57-s + 27·59-s − 15·61-s − 6·67-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 1.13·7-s + 0.353·8-s + 0.904·11-s − 0.832·13-s − 0.727·17-s − 2.06·19-s + 1.96·21-s − 0.625·23-s − 0.612·24-s + 2.30·27-s − 3.34·29-s + 0.538·31-s − 1.56·33-s + 1.97·37-s + 1.44·39-s − 3.27·41-s − 2.74·43-s + 1.31·47-s + 3·49-s + 1.26·51-s − 1.23·53-s − 0.400·56-s + 3.57·57-s + 3.51·59-s − 1.92·61-s − 0.733·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2054956989\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2054956989\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T^{3} + T^{6} \) |
| 5 | \( 1 \) |
| 19 | \( 1 + 9 T - 179 T^{3} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| good | 3 | \( 1 + p T + p^{2} T^{2} + 5 p T^{3} + 4 p^{2} T^{4} + 2 p^{3} T^{5} + 13 p^{2} T^{6} + 2 p^{4} T^{7} + 4 p^{4} T^{8} + 5 p^{4} T^{9} + p^{6} T^{10} + p^{6} T^{11} + p^{6} T^{12} \) |
| 7 | \( 1 + 3 T - 12 T^{2} - 19 T^{3} + 171 T^{4} + 18 p T^{5} - 1161 T^{6} + 18 p^{2} T^{7} + 171 p^{2} T^{8} - 19 p^{3} T^{9} - 12 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 11 | \( ( 1 - T - 10 T^{2} - p T^{3} + p^{2} T^{4} )^{3} \) |
| 13 | \( 1 + 3 T + 12 T^{2} + 58 T^{3} + 288 T^{4} + 981 T^{5} + 2679 T^{6} + 981 p T^{7} + 288 p^{2} T^{8} + 58 p^{3} T^{9} + 12 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 17 | \( 1 + 3 T + 18 T^{2} + 126 T^{3} + 621 T^{4} + 2217 T^{5} + 10549 T^{6} + 2217 p T^{7} + 621 p^{2} T^{8} + 126 p^{3} T^{9} + 18 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 23 | \( 1 + 3 T - 12 T^{2} - 208 T^{3} - 678 T^{4} + 2313 T^{5} + 28901 T^{6} + 2313 p T^{7} - 678 p^{2} T^{8} - 208 p^{3} T^{9} - 12 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 29 | \( 1 + 18 T + 144 T^{2} + 720 T^{3} + 3636 T^{4} + 23976 T^{5} + 146791 T^{6} + 23976 p T^{7} + 3636 p^{2} T^{8} + 720 p^{3} T^{9} + 144 p^{4} T^{10} + 18 p^{5} T^{11} + p^{6} T^{12} \) |
| 31 | \( 1 - 3 T - 75 T^{2} + 82 T^{3} + 3951 T^{4} - 1287 T^{5} - 139914 T^{6} - 1287 p T^{7} + 3951 p^{2} T^{8} + 82 p^{3} T^{9} - 75 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 37 | \( ( 1 - 6 T + 96 T^{2} - 425 T^{3} + 96 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 41 | \( 1 + 21 T + 216 T^{2} + 1620 T^{3} + 10035 T^{4} + 50259 T^{5} + 263737 T^{6} + 50259 p T^{7} + 10035 p^{2} T^{8} + 1620 p^{3} T^{9} + 216 p^{4} T^{10} + 21 p^{5} T^{11} + p^{6} T^{12} \) |
| 43 | \( 1 + 18 T + 252 T^{2} + 2583 T^{3} + 24849 T^{4} + 192573 T^{5} + 1378925 T^{6} + 192573 p T^{7} + 24849 p^{2} T^{8} + 2583 p^{3} T^{9} + 252 p^{4} T^{10} + 18 p^{5} T^{11} + p^{6} T^{12} \) |
| 47 | \( 1 - 9 T + 9 T^{2} + 495 T^{3} - 3222 T^{4} - 3726 T^{5} + 123409 T^{6} - 3726 p T^{7} - 3222 p^{2} T^{8} + 495 p^{3} T^{9} + 9 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \) |
| 53 | \( 1 + 9 T + 162 T^{2} + 1156 T^{3} + 14661 T^{4} + 90351 T^{5} + 897425 T^{6} + 90351 p T^{7} + 14661 p^{2} T^{8} + 1156 p^{3} T^{9} + 162 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \) |
| 59 | \( 1 - 27 T + 360 T^{2} - 3456 T^{3} + 26370 T^{4} - 162423 T^{5} + 1050877 T^{6} - 162423 p T^{7} + 26370 p^{2} T^{8} - 3456 p^{3} T^{9} + 360 p^{4} T^{10} - 27 p^{5} T^{11} + p^{6} T^{12} \) |
| 61 | \( 1 + 15 T + 126 T^{2} + 474 T^{3} + 153 T^{4} - 50871 T^{5} - 528355 T^{6} - 50871 p T^{7} + 153 p^{2} T^{8} + 474 p^{3} T^{9} + 126 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \) |
| 67 | \( 1 + 6 T + 18 T^{2} + 231 T^{3} + 639 T^{4} - 11631 T^{5} - 297739 T^{6} - 11631 p T^{7} + 639 p^{2} T^{8} + 231 p^{3} T^{9} + 18 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 71 | \( 1 - 6 T - 180 T^{3} - 4032 T^{4} + 26598 T^{5} + 151057 T^{6} + 26598 p T^{7} - 4032 p^{2} T^{8} - 180 p^{3} T^{9} - 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 73 | \( 1 + 21 T + 192 T^{2} + 1066 T^{3} - 828 T^{4} - 124155 T^{5} - 1559649 T^{6} - 124155 p T^{7} - 828 p^{2} T^{8} + 1066 p^{3} T^{9} + 192 p^{4} T^{10} + 21 p^{5} T^{11} + p^{6} T^{12} \) |
| 79 | \( 1 - 12 T + 87 T^{2} + 523 T^{3} - 1377 T^{4} - 8181 T^{5} + 778194 T^{6} - 8181 p T^{7} - 1377 p^{2} T^{8} + 523 p^{3} T^{9} + 87 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \) |
| 83 | \( 1 - 18 T + 186 T^{2} - 542 T^{3} - 9888 T^{4} + 154344 T^{5} - 1726273 T^{6} + 154344 p T^{7} - 9888 p^{2} T^{8} - 542 p^{3} T^{9} + 186 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \) |
| 89 | \( 1 + 288 T^{2} - 724 T^{3} + 43056 T^{4} - 167004 T^{5} + 4340087 T^{6} - 167004 p T^{7} + 43056 p^{2} T^{8} - 724 p^{3} T^{9} + 288 p^{4} T^{10} + p^{6} T^{12} \) |
| 97 | \( 1 - 45 T + 828 T^{2} - 7090 T^{3} - 2484 T^{4} + 893997 T^{5} - 12645213 T^{6} + 893997 p T^{7} - 2484 p^{2} T^{8} - 7090 p^{3} T^{9} + 828 p^{4} T^{10} - 45 p^{5} T^{11} + p^{6} T^{12} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.52024215346842317463496907493, −5.02193843484980820866488481704, −4.90527476825322559656945868945, −4.89610199954287676935309577419, −4.87487917966366282707713998532, −4.54398587251902455482778070930, −4.23284456754973742710242467629, −4.22709543072097420660254242717, −4.13606692260187767326310383859, −3.83312796812927014292723683593, −3.59531618777238726149624448908, −3.43921966784764158192626973328, −3.40976056268112993770167189256, −3.05190313532651448053732997255, −2.85543051393648254979955947860, −2.82103807158774387208710951923, −2.23707688591666254237885515290, −2.11505230238034545182611705298, −2.06571362750334865046590796530, −1.90930456979630832457809571251, −1.69141074002638302831368510357, −1.11089051487234001183985372968, −0.856265623157924132484651765759, −0.33677539817380791883586141223, −0.20445843364861855879019466836,
0.20445843364861855879019466836, 0.33677539817380791883586141223, 0.856265623157924132484651765759, 1.11089051487234001183985372968, 1.69141074002638302831368510357, 1.90930456979630832457809571251, 2.06571362750334865046590796530, 2.11505230238034545182611705298, 2.23707688591666254237885515290, 2.82103807158774387208710951923, 2.85543051393648254979955947860, 3.05190313532651448053732997255, 3.40976056268112993770167189256, 3.43921966784764158192626973328, 3.59531618777238726149624448908, 3.83312796812927014292723683593, 4.13606692260187767326310383859, 4.22709543072097420660254242717, 4.23284456754973742710242467629, 4.54398587251902455482778070930, 4.87487917966366282707713998532, 4.89610199954287676935309577419, 4.90527476825322559656945868945, 5.02193843484980820866488481704, 5.52024215346842317463496907493
Plot not available for L-functions of degree greater than 10.
