| L(s) = 1 | − 12·9-s − 96·25-s − 96·29-s + 192·41-s − 80·53-s + 192·61-s + 576·73-s + 90·81-s + 192·89-s + 384·97-s + 960·101-s − 192·109-s + 112·113-s + 584·121-s + 192·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 384·169-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | − 4/3·9-s − 3.83·25-s − 3.31·29-s + 4.68·41-s − 1.50·53-s + 3.14·61-s + 7.89·73-s + 10/9·81-s + 2.15·89-s + 3.95·97-s + 9.50·101-s − 1.76·109-s + 0.991·113-s + 4.82·121-s + 1.53·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 2.27·169-s + 0.00578·173-s + 0.00558·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(29.78143414\) |
| \(L(\frac12)\) |
\(\approx\) |
\(29.78143414\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 + p T^{2} )^{4} \) |
| 7 | \( 1 \) |
| good | 5 | \( ( 1 + 48 T^{2} - 96 T^{3} + 1346 T^{4} - 96 p^{2} T^{5} + 48 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 11 | \( 1 - 584 T^{2} + 167452 T^{4} - 31553912 T^{6} + 4373265670 T^{8} - 31553912 p^{4} T^{10} + 167452 p^{8} T^{12} - 584 p^{12} T^{14} + p^{16} T^{16} \) |
| 13 | \( ( 1 + 192 T^{2} + 1344 T^{3} + 7970 T^{4} + 1344 p^{2} T^{5} + 192 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 17 | \( ( 1 + 528 T^{2} + 672 T^{3} + 220130 T^{4} + 672 p^{2} T^{5} + 528 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 19 | \( 1 - 1928 T^{2} + 1783900 T^{4} - 1059171896 T^{6} + 447269664262 T^{8} - 1059171896 p^{4} T^{10} + 1783900 p^{8} T^{12} - 1928 p^{12} T^{14} + p^{16} T^{16} \) |
| 23 | \( 1 - 1928 T^{2} + 1932124 T^{4} - 1454834360 T^{6} + 872243294278 T^{8} - 1454834360 p^{4} T^{10} + 1932124 p^{8} T^{12} - 1928 p^{12} T^{14} + p^{16} T^{16} \) |
| 29 | \( ( 1 + 48 T + 3556 T^{2} + 114192 T^{3} + 4574694 T^{4} + 114192 p^{2} T^{5} + 3556 p^{4} T^{6} + 48 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 31 | \( 1 - 4232 T^{2} + 9689116 T^{4} - 14891091896 T^{6} + 16606329017926 T^{8} - 14891091896 p^{4} T^{10} + 9689116 p^{8} T^{12} - 4232 p^{12} T^{14} + p^{16} T^{16} \) |
| 37 | \( ( 1 + 1636 T^{2} - 105984 T^{3} + 99750 T^{4} - 105984 p^{2} T^{5} + 1636 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 41 | \( ( 1 - 96 T + 7536 T^{2} - 422016 T^{3} + 19400354 T^{4} - 422016 p^{2} T^{5} + 7536 p^{4} T^{6} - 96 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 43 | \( 1 - 5576 T^{2} + 20016988 T^{4} - 46850763512 T^{6} + 96062127228550 T^{8} - 46850763512 p^{4} T^{10} + 20016988 p^{8} T^{12} - 5576 p^{12} T^{14} + p^{16} T^{16} \) |
| 47 | \( 1 - 8456 T^{2} + 28082716 T^{4} - 44170889528 T^{6} + 55756301244742 T^{8} - 44170889528 p^{4} T^{10} + 28082716 p^{8} T^{12} - 8456 p^{12} T^{14} + p^{16} T^{16} \) |
| 53 | \( ( 1 + 40 T + 7804 T^{2} + 343384 T^{3} + 29078758 T^{4} + 343384 p^{2} T^{5} + 7804 p^{4} T^{6} + 40 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 59 | \( 1 - 21896 T^{2} + 224587228 T^{4} - 1409309727032 T^{6} + 5922457985328262 T^{8} - 1409309727032 p^{4} T^{10} + 224587228 p^{8} T^{12} - 21896 p^{12} T^{14} + p^{16} T^{16} \) |
| 61 | \( ( 1 - 96 T + 17184 T^{2} - 1068960 T^{3} + 99806114 T^{4} - 1068960 p^{2} T^{5} + 17184 p^{4} T^{6} - 96 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 67 | \( 1 - 23240 T^{2} + 238231516 T^{4} - 1505735752952 T^{6} + 7309602668695558 T^{8} - 1505735752952 p^{4} T^{10} + 238231516 p^{8} T^{12} - 23240 p^{12} T^{14} + p^{16} T^{16} \) |
| 71 | \( 1 - 17288 T^{2} + 123269212 T^{4} - 392735012024 T^{6} + 943741770326854 T^{8} - 392735012024 p^{4} T^{10} + 123269212 p^{8} T^{12} - 17288 p^{12} T^{14} + p^{16} T^{16} \) |
| 73 | \( ( 1 - 288 T + 49248 T^{2} - 5605536 T^{3} + 476605250 T^{4} - 5605536 p^{2} T^{5} + 49248 p^{4} T^{6} - 288 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 79 | \( 1 - 19976 T^{2} + 232744732 T^{4} - 1840641279032 T^{6} + 12411054418089286 T^{8} - 1840641279032 p^{4} T^{10} + 232744732 p^{8} T^{12} - 19976 p^{12} T^{14} + p^{16} T^{16} \) |
| 83 | \( 1 - 37256 T^{2} + 681767260 T^{4} - 7971195138104 T^{6} + 65017943250589702 T^{8} - 7971195138104 p^{4} T^{10} + 681767260 p^{8} T^{12} - 37256 p^{12} T^{14} + p^{16} T^{16} \) |
| 89 | \( ( 1 - 96 T + 33456 T^{2} - 2237760 T^{3} + 403501346 T^{4} - 2237760 p^{2} T^{5} + 33456 p^{4} T^{6} - 96 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 97 | \( ( 1 - 192 T + 30432 T^{2} - 2692416 T^{3} + 3014210 p T^{4} - 2692416 p^{2} T^{5} + 30432 p^{4} T^{6} - 192 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.64880029928680608782700377955, −3.39615860166692887511608620304, −3.30986102464511793561902996558, −3.21558094987473876711467568864, −3.09557020171902805938543955196, −3.03269986331018381015646109496, −2.72478385701659304233167680092, −2.69688900160073956411470560698, −2.66376295562903167669375183354, −2.16659676192135614986374084410, −2.07000927447407444242821554508, −2.04722645348852821068890387274, −2.00039356103255187104433080732, −1.96739742049262343853521688450, −1.96070907688121460273594524190, −1.85855353782703901230237722345, −1.81896835282098436830233321161, −1.15434780896596864644228963229, −0.894171602668434081322755763307, −0.833693114197861848666315208374, −0.76674633745475576730830955704, −0.63738498397711260233032480598, −0.43397897667489969081823546134, −0.41755269717380542877983633510, −0.36754070934774499711056509059,
0.36754070934774499711056509059, 0.41755269717380542877983633510, 0.43397897667489969081823546134, 0.63738498397711260233032480598, 0.76674633745475576730830955704, 0.833693114197861848666315208374, 0.894171602668434081322755763307, 1.15434780896596864644228963229, 1.81896835282098436830233321161, 1.85855353782703901230237722345, 1.96070907688121460273594524190, 1.96739742049262343853521688450, 2.00039356103255187104433080732, 2.04722645348852821068890387274, 2.07000927447407444242821554508, 2.16659676192135614986374084410, 2.66376295562903167669375183354, 2.69688900160073956411470560698, 2.72478385701659304233167680092, 3.03269986331018381015646109496, 3.09557020171902805938543955196, 3.21558094987473876711467568864, 3.30986102464511793561902996558, 3.39615860166692887511608620304, 3.64880029928680608782700377955
Plot not available for L-functions of degree greater than 10.
