| L(s) = 1 | − 32·2-s + 1.02e3·4-s + 6.45e3·5-s − 1.68e4·7-s − 3.27e4·8-s − 2.06e5·10-s − 8.03e4·11-s + 1.91e6·13-s + 5.37e5·14-s + 1.04e6·16-s − 1.13e7·17-s + 9.87e6·19-s + 6.60e6·20-s + 2.57e6·22-s + 1.32e7·23-s − 7.22e6·25-s − 6.14e7·26-s − 1.72e7·28-s − 2.03e8·29-s − 2.81e8·31-s − 3.35e7·32-s + 3.63e8·34-s − 1.08e8·35-s + 4.11e8·37-s − 3.15e8·38-s − 2.11e8·40-s + 8.12e8·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.923·5-s − 0.377·7-s − 0.353·8-s − 0.652·10-s − 0.150·11-s + 1.43·13-s + 0.267·14-s + 0.250·16-s − 1.93·17-s + 0.914·19-s + 0.461·20-s + 0.106·22-s + 0.429·23-s − 0.147·25-s − 1.01·26-s − 0.188·28-s − 1.84·29-s − 1.76·31-s − 0.176·32-s + 1.37·34-s − 0.348·35-s + 0.975·37-s − 0.646·38-s − 0.326·40-s + 1.09·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 32T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + 1.68e4T \) |
| good | 5 | \( 1 - 6.45e3T + 4.88e7T^{2} \) |
| 11 | \( 1 + 8.03e4T + 2.85e11T^{2} \) |
| 13 | \( 1 - 1.91e6T + 1.79e12T^{2} \) |
| 17 | \( 1 + 1.13e7T + 3.42e13T^{2} \) |
| 19 | \( 1 - 9.87e6T + 1.16e14T^{2} \) |
| 23 | \( 1 - 1.32e7T + 9.52e14T^{2} \) |
| 29 | \( 1 + 2.03e8T + 1.22e16T^{2} \) |
| 31 | \( 1 + 2.81e8T + 2.54e16T^{2} \) |
| 37 | \( 1 - 4.11e8T + 1.77e17T^{2} \) |
| 41 | \( 1 - 8.12e8T + 5.50e17T^{2} \) |
| 43 | \( 1 - 5.98e8T + 9.29e17T^{2} \) |
| 47 | \( 1 + 6.96e8T + 2.47e18T^{2} \) |
| 53 | \( 1 + 5.96e8T + 9.26e18T^{2} \) |
| 59 | \( 1 + 3.86e9T + 3.01e19T^{2} \) |
| 61 | \( 1 - 1.08e10T + 4.35e19T^{2} \) |
| 67 | \( 1 + 1.03e10T + 1.22e20T^{2} \) |
| 71 | \( 1 + 2.04e9T + 2.31e20T^{2} \) |
| 73 | \( 1 + 3.07e10T + 3.13e20T^{2} \) |
| 79 | \( 1 - 1.71e10T + 7.47e20T^{2} \) |
| 83 | \( 1 - 6.60e10T + 1.28e21T^{2} \) |
| 89 | \( 1 - 4.73e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + 1.15e11T + 7.15e21T^{2} \) |
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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
−10.77168821956056973129306037220, −9.381813158267832565822248897840, −9.025320303970116455191354324869, −7.56314877141525932657100999231, −6.40848614736702262235760194834, −5.57959507547576190166960684871, −3.81345036295412078587125622926, −2.38242533401950839501280304033, −1.38170975612034733685096708291, 0,
1.38170975612034733685096708291, 2.38242533401950839501280304033, 3.81345036295412078587125622926, 5.57959507547576190166960684871, 6.40848614736702262235760194834, 7.56314877141525932657100999231, 9.025320303970116455191354324869, 9.381813158267832565822248897840, 10.77168821956056973129306037220
