| L(s) = 1 | − 32·2-s + 738·3-s + 1.02e3·4-s − 3.12e3·5-s − 2.36e4·6-s + 2.55e4·7-s − 3.27e4·8-s + 3.67e5·9-s + 1.00e5·10-s + 7.69e5·11-s + 7.55e5·12-s − 9.18e5·13-s − 8.18e5·14-s − 2.30e6·15-s + 1.04e6·16-s + 1.03e7·17-s − 1.17e7·18-s − 5.52e6·19-s − 3.20e6·20-s + 1.88e7·21-s − 2.46e7·22-s − 3.99e7·23-s − 2.41e7·24-s + 9.76e6·25-s + 2.94e7·26-s + 1.40e8·27-s + 2.61e7·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.75·3-s + 1/2·4-s − 0.447·5-s − 1.23·6-s + 0.575·7-s − 0.353·8-s + 2.07·9-s + 0.316·10-s + 1.43·11-s + 0.876·12-s − 0.686·13-s − 0.406·14-s − 0.784·15-s + 1/4·16-s + 1.76·17-s − 1.46·18-s − 0.511·19-s − 0.223·20-s + 1.00·21-s − 1.01·22-s − 1.29·23-s − 0.619·24-s + 1/5·25-s + 0.485·26-s + 1.88·27-s + 0.287·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(2.145046318\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.145046318\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{5} T \) |
| 5 | \( 1 + p^{5} T \) |
| good | 3 | \( 1 - 82 p^{2} T + p^{11} T^{2} \) |
| 7 | \( 1 - 25574 T + p^{11} T^{2} \) |
| 11 | \( 1 - 769152 T + p^{11} T^{2} \) |
| 13 | \( 1 + 918982 T + p^{11} T^{2} \) |
| 17 | \( 1 - 10312794 T + p^{11} T^{2} \) |
| 19 | \( 1 + 5521660 T + p^{11} T^{2} \) |
| 23 | \( 1 + 39973422 T + p^{11} T^{2} \) |
| 29 | \( 1 + 15269010 T + p^{11} T^{2} \) |
| 31 | \( 1 + 241583788 T + p^{11} T^{2} \) |
| 37 | \( 1 + 25751446 T + p^{11} T^{2} \) |
| 41 | \( 1 + 1217700138 T + p^{11} T^{2} \) |
| 43 | \( 1 + 683436262 T + p^{11} T^{2} \) |
| 47 | \( 1 - 1537395294 T + p^{11} T^{2} \) |
| 53 | \( 1 - 3572891298 T + p^{11} T^{2} \) |
| 59 | \( 1 + 1069039020 T + p^{11} T^{2} \) |
| 61 | \( 1 + 2091535078 T + p^{11} T^{2} \) |
| 67 | \( 1 + 1462369186 T + p^{11} T^{2} \) |
| 71 | \( 1 - 9660178332 T + p^{11} T^{2} \) |
| 73 | \( 1 + 5603447662 T + p^{11} T^{2} \) |
| 79 | \( 1 - 5026936280 T + p^{11} T^{2} \) |
| 83 | \( 1 + 38405955462 T + p^{11} T^{2} \) |
| 89 | \( 1 - 35558583210 T + p^{11} T^{2} \) |
| 97 | \( 1 - 10572232514 T + p^{11} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−18.59395154180480337361259153007, −16.71703116852337670862261386233, −14.96347327723118778655366335783, −14.25155787275088476256690506691, −12.09786580101310771456610274033, −9.850292064835279811918489614871, −8.555893328828761291808658869184, −7.42697835426948188139130996525, −3.65484531223185863483635625945, −1.71072846945230059690854602385,
1.71072846945230059690854602385, 3.65484531223185863483635625945, 7.42697835426948188139130996525, 8.555893328828761291808658869184, 9.850292064835279811918489614871, 12.09786580101310771456610274033, 14.25155787275088476256690506691, 14.96347327723118778655366335783, 16.71703116852337670862261386233, 18.59395154180480337361259153007
