| L(s) = 1 | + 2-s + 3-s + 6-s − 4·7-s + 5·8-s + 3·9-s − 10·11-s − 15·13-s − 4·14-s + 8·16-s + 17-s + 3·18-s − 4·21-s − 10·22-s + 4·23-s + 5·24-s − 15·26-s + 10·27-s + 2·29-s − 2·31-s + 4·32-s − 10·33-s + 34-s + 4·37-s − 15·39-s + 2·41-s − 4·42-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.408·6-s − 1.51·7-s + 1.76·8-s + 9-s − 3.01·11-s − 4.16·13-s − 1.06·14-s + 2·16-s + 0.242·17-s + 0.707·18-s − 0.872·21-s − 2.13·22-s + 0.834·23-s + 1.02·24-s − 2.94·26-s + 1.92·27-s + 0.371·29-s − 0.359·31-s + 0.707·32-s − 1.74·33-s + 0.171·34-s + 0.657·37-s − 2.40·39-s + 0.312·41-s − 0.617·42-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2383379521\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2383379521\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 19 | \( 1 + 7 p T^{3} + p^{3} T^{6} \) |
| good | 2 | \( 1 - T + T^{2} - 3 p T^{3} + 3 T^{4} - p^{2} T^{5} + 21 T^{6} - p^{3} T^{7} + 3 p^{2} T^{8} - 3 p^{4} T^{9} + p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \) |
| 3 | \( 1 - T - 2 T^{2} - 5 T^{3} + T^{4} + 4 p T^{5} + 19 T^{6} + 4 p^{2} T^{7} + p^{2} T^{8} - 5 p^{3} T^{9} - 2 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \) |
| 7 | \( ( 1 + 2 T + 16 T^{2} + 29 T^{3} + 16 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 11 | \( ( 1 + 5 T + 35 T^{2} + 109 T^{3} + 35 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 13 | \( ( 1 - 2 T + p T^{2} )^{3}( 1 + 7 T + p T^{2} )^{3} \) |
| 17 | \( 1 - T - 6 T^{2} + 47 T^{3} - 97 T^{4} - 240 T^{5} + 9433 T^{6} - 240 p T^{7} - 97 p^{2} T^{8} + 47 p^{3} T^{9} - 6 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \) |
| 23 | \( 1 - 4 T - 14 T^{2} + 150 T^{3} - 330 T^{4} - 646 T^{5} + 13395 T^{6} - 646 p T^{7} - 330 p^{2} T^{8} + 150 p^{3} T^{9} - 14 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \) |
| 29 | \( 1 - 2 T - 78 T^{2} + 70 T^{3} + 4112 T^{4} - 1764 T^{5} - 135893 T^{6} - 1764 p T^{7} + 4112 p^{2} T^{8} + 70 p^{3} T^{9} - 78 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \) |
| 31 | \( ( 1 + T + 87 T^{2} + 55 T^{3} + 87 p T^{4} + p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 37 | \( ( 1 - 2 T - 8 T^{2} + 79 T^{3} - 8 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 41 | \( 1 - 2 T - 76 T^{2} + 94 T^{3} + 2866 T^{4} - 402 T^{5} - 115153 T^{6} - 402 p T^{7} + 2866 p^{2} T^{8} + 94 p^{3} T^{9} - 76 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \) |
| 43 | \( 1 + T - 84 T^{2} + 155 T^{3} + 3605 T^{4} - 8436 T^{5} - 152285 T^{6} - 8436 p T^{7} + 3605 p^{2} T^{8} + 155 p^{3} T^{9} - 84 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \) |
| 47 | \( 1 - 6 T - 98 T^{2} + 226 T^{3} + 8568 T^{4} - 6688 T^{5} - 450585 T^{6} - 6688 p T^{7} + 8568 p^{2} T^{8} + 226 p^{3} T^{9} - 98 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 53 | \( 1 - 11 T + 4 T^{2} + 423 T^{3} - 1917 T^{4} + 5488 T^{5} - 36627 T^{6} + 5488 p T^{7} - 1917 p^{2} T^{8} + 423 p^{3} T^{9} + 4 p^{4} T^{10} - 11 p^{5} T^{11} + p^{6} T^{12} \) |
| 59 | \( 1 + 6 T - 134 T^{2} - 298 T^{3} + 14916 T^{4} + 12556 T^{5} - 985377 T^{6} + 12556 p T^{7} + 14916 p^{2} T^{8} - 298 p^{3} T^{9} - 134 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 61 | \( 1 - 9 T - 53 T^{2} + 892 T^{3} + 341 T^{4} - 26923 T^{5} + 157646 T^{6} - 26923 p T^{7} + 341 p^{2} T^{8} + 892 p^{3} T^{9} - 53 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \) |
| 67 | \( 1 + 20 T + 91 T^{2} + 644 T^{3} + 19418 T^{4} + 116972 T^{5} + 70523 T^{6} + 116972 p T^{7} + 19418 p^{2} T^{8} + 644 p^{3} T^{9} + 91 p^{4} T^{10} + 20 p^{5} T^{11} + p^{6} T^{12} \) |
| 71 | \( 1 - 29 T + 392 T^{2} - 3851 T^{3} + 36757 T^{4} - 355872 T^{5} + 3184895 T^{6} - 355872 p T^{7} + 36757 p^{2} T^{8} - 3851 p^{3} T^{9} + 392 p^{4} T^{10} - 29 p^{5} T^{11} + p^{6} T^{12} \) |
| 73 | \( 1 + 22 T + 148 T^{2} + 814 T^{3} + 16010 T^{4} + 143110 T^{5} + 774911 T^{6} + 143110 p T^{7} + 16010 p^{2} T^{8} + 814 p^{3} T^{9} + 148 p^{4} T^{10} + 22 p^{5} T^{11} + p^{6} T^{12} \) |
| 79 | \( 1 - 24 T + 223 T^{2} - 1384 T^{3} + 11666 T^{4} - 62240 T^{5} + 107087 T^{6} - 62240 p T^{7} + 11666 p^{2} T^{8} - 1384 p^{3} T^{9} + 223 p^{4} T^{10} - 24 p^{5} T^{11} + p^{6} T^{12} \) |
| 83 | \( ( 1 - 3 T + 195 T^{2} - 575 T^{3} + 195 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 89 | \( 1 - 14 T - 35 T^{2} + 1638 T^{3} - 1302 T^{4} - 97958 T^{5} + 1042389 T^{6} - 97958 p T^{7} - 1302 p^{2} T^{8} + 1638 p^{3} T^{9} - 35 p^{4} T^{10} - 14 p^{5} T^{11} + p^{6} T^{12} \) |
| 97 | \( 1 - 7 T - 176 T^{2} + 899 T^{3} + 20141 T^{4} - 38638 T^{5} - 2016151 T^{6} - 38638 p T^{7} + 20141 p^{2} T^{8} + 899 p^{3} T^{9} - 176 p^{4} T^{10} - 7 p^{5} T^{11} + p^{6} T^{12} \) |
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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
−6.14872634281463052378911103775, −5.49100037982058781935466097656, −5.35146348392470295361451525187, −5.27241373098238115341083553711, −5.19059935917495014728301096912, −5.02710603811206847707130621460, −4.78987977491974463965324830109, −4.77901647395672804896443977754, −4.59389301000982668979591950169, −4.56386455995862912981249748125, −4.18516220977094387832365994037, −3.79261435691656696779702123729, −3.67358604037290854915818680371, −3.53184020583659957754294386384, −3.40688307374145805359966193421, −2.80181047719129562755059651019, −2.72906864628016150800411087153, −2.68650354244794469068216419111, −2.46877666432954634516220727658, −2.45494575670366612903584550670, −2.09982137798414405335413691955, −1.54501951332436919256316370814, −1.37590370879508833567984685784, −0.851493773775849930987144135948, −0.10201020570546821500503383326,
0.10201020570546821500503383326, 0.851493773775849930987144135948, 1.37590370879508833567984685784, 1.54501951332436919256316370814, 2.09982137798414405335413691955, 2.45494575670366612903584550670, 2.46877666432954634516220727658, 2.68650354244794469068216419111, 2.72906864628016150800411087153, 2.80181047719129562755059651019, 3.40688307374145805359966193421, 3.53184020583659957754294386384, 3.67358604037290854915818680371, 3.79261435691656696779702123729, 4.18516220977094387832365994037, 4.56386455995862912981249748125, 4.59389301000982668979591950169, 4.77901647395672804896443977754, 4.78987977491974463965324830109, 5.02710603811206847707130621460, 5.19059935917495014728301096912, 5.27241373098238115341083553711, 5.35146348392470295361451525187, 5.49100037982058781935466097656, 6.14872634281463052378911103775
Plot not available for L-functions of degree greater than 10.
