| L(s) = 1 | + 2.09e6·2-s − 5.44e9·3-s + 4.39e12·4-s − 1.56e15·5-s − 1.14e16·6-s + 5.58e17·7-s + 9.22e18·8-s − 2.98e20·9-s − 3.27e21·10-s + 5.35e21·11-s − 2.39e22·12-s + 3.95e23·13-s + 1.17e24·14-s + 8.50e24·15-s + 1.93e25·16-s − 5.26e26·17-s − 6.26e26·18-s + 1.48e27·19-s − 6.86e27·20-s − 3.04e27·21-s + 1.12e28·22-s − 9.08e28·23-s − 5.02e28·24-s + 1.29e30·25-s + 8.29e29·26-s + 3.41e30·27-s + 2.45e30·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.300·3-s + 0.5·4-s − 1.46·5-s − 0.212·6-s + 0.377·7-s + 0.353·8-s − 0.909·9-s − 1.03·10-s + 0.217·11-s − 0.150·12-s + 0.443·13-s + 0.267·14-s + 0.440·15-s + 0.250·16-s − 1.84·17-s − 0.643·18-s + 0.477·19-s − 0.731·20-s − 0.113·21-s + 0.154·22-s − 0.479·23-s − 0.106·24-s + 1.14·25-s + 0.313·26-s + 0.574·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(44-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+43/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(22)\) |
\(\approx\) |
\(1.249910100\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.249910100\) |
| \(L(\frac{45}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2.09e6T \) |
| 7 | \( 1 - 5.58e17T \) |
| good | 3 | \( 1 + 5.44e9T + 3.28e20T^{2} \) |
| 5 | \( 1 + 1.56e15T + 1.13e30T^{2} \) |
| 11 | \( 1 - 5.35e21T + 6.02e44T^{2} \) |
| 13 | \( 1 - 3.95e23T + 7.93e47T^{2} \) |
| 17 | \( 1 + 5.26e26T + 8.11e52T^{2} \) |
| 19 | \( 1 - 1.48e27T + 9.69e54T^{2} \) |
| 23 | \( 1 + 9.08e28T + 3.58e58T^{2} \) |
| 29 | \( 1 + 3.65e31T + 7.64e62T^{2} \) |
| 31 | \( 1 + 1.37e32T + 1.34e64T^{2} \) |
| 37 | \( 1 - 4.35e32T + 2.70e67T^{2} \) |
| 41 | \( 1 + 5.75e34T + 2.23e69T^{2} \) |
| 43 | \( 1 - 5.90e34T + 1.73e70T^{2} \) |
| 47 | \( 1 - 5.17e35T + 7.94e71T^{2} \) |
| 53 | \( 1 + 4.68e36T + 1.39e74T^{2} \) |
| 59 | \( 1 + 1.38e38T + 1.40e76T^{2} \) |
| 61 | \( 1 - 3.03e37T + 5.87e76T^{2} \) |
| 67 | \( 1 - 3.07e39T + 3.32e78T^{2} \) |
| 71 | \( 1 - 5.51e39T + 4.01e79T^{2} \) |
| 73 | \( 1 - 2.24e40T + 1.32e80T^{2} \) |
| 79 | \( 1 + 3.95e40T + 3.96e81T^{2} \) |
| 83 | \( 1 + 1.32e41T + 3.31e82T^{2} \) |
| 89 | \( 1 - 7.96e40T + 6.66e83T^{2} \) |
| 97 | \( 1 - 2.39e42T + 2.69e85T^{2} \) |
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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
−11.33000739392422551257195533908, −11.13309887122886748074422521390, −8.879020676949777977420572756919, −7.80460488426936710619498562007, −6.65223747170802835623383155585, −5.35505394657838227133191780497, −4.22471188701861997629135200282, −3.42131008284811201782985330430, −2.02628766132099438966533973520, −0.42552477466846300073813353671,
0.42552477466846300073813353671, 2.02628766132099438966533973520, 3.42131008284811201782985330430, 4.22471188701861997629135200282, 5.35505394657838227133191780497, 6.65223747170802835623383155585, 7.80460488426936710619498562007, 8.879020676949777977420572756919, 11.13309887122886748074422521390, 11.33000739392422551257195533908
