| L(s) = 1 | + 2.09e6·2-s + 1.06e10·3-s + 4.39e12·4-s − 4.44e14·5-s + 2.22e16·6-s − 2.02e18·7-s + 9.22e18·8-s − 2.15e20·9-s − 9.31e20·10-s − 8.57e21·11-s + 4.66e22·12-s − 7.14e23·13-s − 4.24e24·14-s − 4.71e24·15-s + 1.93e25·16-s + 2.68e26·17-s − 4.51e26·18-s + 6.14e26·19-s − 1.95e27·20-s − 2.15e28·21-s − 1.79e28·22-s − 2.05e29·23-s + 9.79e28·24-s − 9.39e29·25-s − 1.49e30·26-s − 5.77e30·27-s − 8.90e30·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.586·3-s + 0.5·4-s − 0.416·5-s + 0.414·6-s − 1.37·7-s + 0.353·8-s − 0.656·9-s − 0.294·10-s − 0.349·11-s + 0.293·12-s − 0.802·13-s − 0.969·14-s − 0.244·15-s + 0.250·16-s + 0.942·17-s − 0.464·18-s + 0.197·19-s − 0.208·20-s − 0.803·21-s − 0.247·22-s − 1.08·23-s + 0.207·24-s − 0.826·25-s − 0.567·26-s − 0.970·27-s − 0.685·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(44-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s+43/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(22)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{45}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2.09e6T \) |
| good | 3 | \( 1 - 1.06e10T + 3.28e20T^{2} \) |
| 5 | \( 1 + 4.44e14T + 1.13e30T^{2} \) |
| 7 | \( 1 + 2.02e18T + 2.18e36T^{2} \) |
| 11 | \( 1 + 8.57e21T + 6.02e44T^{2} \) |
| 13 | \( 1 + 7.14e23T + 7.93e47T^{2} \) |
| 17 | \( 1 - 2.68e26T + 8.11e52T^{2} \) |
| 19 | \( 1 - 6.14e26T + 9.69e54T^{2} \) |
| 23 | \( 1 + 2.05e29T + 3.58e58T^{2} \) |
| 29 | \( 1 + 5.24e31T + 7.64e62T^{2} \) |
| 31 | \( 1 - 1.41e32T + 1.34e64T^{2} \) |
| 37 | \( 1 - 8.30e33T + 2.70e67T^{2} \) |
| 41 | \( 1 + 6.87e33T + 2.23e69T^{2} \) |
| 43 | \( 1 + 1.85e35T + 1.73e70T^{2} \) |
| 47 | \( 1 - 1.55e36T + 7.94e71T^{2} \) |
| 53 | \( 1 - 6.82e36T + 1.39e74T^{2} \) |
| 59 | \( 1 + 3.72e37T + 1.40e76T^{2} \) |
| 61 | \( 1 - 1.67e38T + 5.87e76T^{2} \) |
| 67 | \( 1 + 1.60e39T + 3.32e78T^{2} \) |
| 71 | \( 1 + 1.10e40T + 4.01e79T^{2} \) |
| 73 | \( 1 - 4.41e39T + 1.32e80T^{2} \) |
| 79 | \( 1 - 6.24e40T + 3.96e81T^{2} \) |
| 83 | \( 1 - 1.02e41T + 3.31e82T^{2} \) |
| 89 | \( 1 - 5.28e41T + 6.66e83T^{2} \) |
| 97 | \( 1 + 1.00e43T + 2.69e85T^{2} \) |
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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
−16.53781721056229382766065810149, −15.03833840870441340944184556413, −13.49444893533522421817150316312, −11.98324779812143640646026337515, −9.766156333591056102206689597675, −7.68880914418365397749002701642, −5.83719755484291900652658897519, −3.71575880151204210972812025857, −2.57961558149180729557837707894, 0,
2.57961558149180729557837707894, 3.71575880151204210972812025857, 5.83719755484291900652658897519, 7.68880914418365397749002701642, 9.766156333591056102206689597675, 11.98324779812143640646026337515, 13.49444893533522421817150316312, 15.03833840870441340944184556413, 16.53781721056229382766065810149
