| L(s) = 1 | − 4.29e9·2-s + 3.37e15·3-s + 1.84e19·4-s − 9.92e22·5-s − 1.45e25·6-s + 1.82e27·7-s − 7.92e28·8-s + 1.10e30·9-s + 4.26e32·10-s + 9.77e33·11-s + 6.23e34·12-s + 6.25e35·13-s − 7.82e36·14-s − 3.35e38·15-s + 3.40e38·16-s + 4.66e39·17-s − 4.75e39·18-s − 6.30e41·19-s − 1.83e42·20-s + 6.15e42·21-s − 4.19e43·22-s + 1.20e44·23-s − 2.67e44·24-s + 7.14e45·25-s − 2.68e45·26-s − 3.10e46·27-s + 3.36e46·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.05·3-s + 0.5·4-s − 1.90·5-s − 0.744·6-s + 0.623·7-s − 0.353·8-s + 0.107·9-s + 1.34·10-s + 1.39·11-s + 0.526·12-s + 0.391·13-s − 0.440·14-s − 2.00·15-s + 0.250·16-s + 0.477·17-s − 0.0760·18-s − 1.73·19-s − 0.953·20-s + 0.656·21-s − 0.986·22-s + 0.668·23-s − 0.372·24-s + 2.63·25-s − 0.277·26-s − 0.939·27-s + 0.311·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(66-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s+65/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(33)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{67}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 4.29e9T \) |
| good | 3 | \( 1 - 3.37e15T + 1.03e31T^{2} \) |
| 5 | \( 1 + 9.92e22T + 2.71e45T^{2} \) |
| 7 | \( 1 - 1.82e27T + 8.53e54T^{2} \) |
| 11 | \( 1 - 9.77e33T + 4.90e67T^{2} \) |
| 13 | \( 1 - 6.25e35T + 2.54e72T^{2} \) |
| 17 | \( 1 - 4.66e39T + 9.53e79T^{2} \) |
| 19 | \( 1 + 6.30e41T + 1.31e83T^{2} \) |
| 23 | \( 1 - 1.20e44T + 3.25e88T^{2} \) |
| 29 | \( 1 - 1.55e47T + 1.13e95T^{2} \) |
| 31 | \( 1 - 2.16e45T + 8.67e96T^{2} \) |
| 37 | \( 1 + 1.77e51T + 8.57e101T^{2} \) |
| 41 | \( 1 + 2.46e52T + 6.77e104T^{2} \) |
| 43 | \( 1 - 9.28e52T + 1.49e106T^{2} \) |
| 47 | \( 1 + 1.11e54T + 4.85e108T^{2} \) |
| 53 | \( 1 - 5.65e55T + 1.19e112T^{2} \) |
| 59 | \( 1 + 4.19e56T + 1.27e115T^{2} \) |
| 61 | \( 1 + 1.52e58T + 1.11e116T^{2} \) |
| 67 | \( 1 - 1.88e58T + 4.95e118T^{2} \) |
| 71 | \( 1 + 1.02e59T + 2.14e120T^{2} \) |
| 73 | \( 1 + 1.34e60T + 1.30e121T^{2} \) |
| 79 | \( 1 + 4.82e61T + 2.21e123T^{2} \) |
| 83 | \( 1 + 6.90e61T + 5.49e124T^{2} \) |
| 89 | \( 1 + 2.76e63T + 5.13e126T^{2} \) |
| 97 | \( 1 + 6.69e64T + 1.38e129T^{2} \) |
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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
−14.50585826568165695420702706957, −12.11516262784532923352961072083, −11.00650203861176988462755067699, −8.773650496389907882926035363201, −8.253556057623910300942773880441, −6.95591752821259754277221512982, −4.20552825022655629107139508238, −3.21109506436071256892383807905, −1.47240413047536480591812892851, 0,
1.47240413047536480591812892851, 3.21109506436071256892383807905, 4.20552825022655629107139508238, 6.95591752821259754277221512982, 8.253556057623910300942773880441, 8.773650496389907882926035363201, 11.00650203861176988462755067699, 12.11516262784532923352961072083, 14.50585826568165695420702706957
