| L(s) = 1 | − 6.87e10·2-s − 2.54e17·3-s + 4.72e21·4-s + 3.27e25·5-s + 1.74e28·6-s + 8.99e30·7-s − 3.24e32·8-s − 2.93e33·9-s − 2.25e36·10-s − 1.33e38·11-s − 1.20e39·12-s + 5.38e39·13-s − 6.18e41·14-s − 8.32e42·15-s + 2.23e43·16-s − 4.04e44·17-s + 2.01e44·18-s − 3.68e45·19-s + 1.54e47·20-s − 2.28e48·21-s + 9.17e48·22-s + 9.78e49·23-s + 8.25e49·24-s + 1.44e49·25-s − 3.69e50·26-s + 1.79e52·27-s + 4.24e52·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.978·3-s + 0.5·4-s + 1.00·5-s + 0.691·6-s + 1.28·7-s − 0.353·8-s − 0.0434·9-s − 0.711·10-s − 1.30·11-s − 0.489·12-s + 0.118·13-s − 0.906·14-s − 0.984·15-s + 0.250·16-s − 0.496·17-s + 0.0307·18-s − 0.0780·19-s + 0.503·20-s − 1.25·21-s + 0.921·22-s + 1.93·23-s + 0.345·24-s + 0.0136·25-s − 0.0834·26-s + 1.02·27-s + 0.641·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(74-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s+73/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(37)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{75}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 6.87e10T \) |
| good | 3 | \( 1 + 2.54e17T + 6.75e34T^{2} \) |
| 5 | \( 1 - 3.27e25T + 1.05e51T^{2} \) |
| 7 | \( 1 - 8.99e30T + 4.92e61T^{2} \) |
| 11 | \( 1 + 1.33e38T + 1.05e76T^{2} \) |
| 13 | \( 1 - 5.38e39T + 2.07e81T^{2} \) |
| 17 | \( 1 + 4.04e44T + 6.64e89T^{2} \) |
| 19 | \( 1 + 3.68e45T + 2.23e93T^{2} \) |
| 23 | \( 1 - 9.78e49T + 2.54e99T^{2} \) |
| 29 | \( 1 + 3.94e53T + 5.68e106T^{2} \) |
| 31 | \( 1 + 2.44e54T + 7.40e108T^{2} \) |
| 37 | \( 1 - 1.21e57T + 3.01e114T^{2} \) |
| 41 | \( 1 - 7.27e58T + 5.41e117T^{2} \) |
| 43 | \( 1 + 2.70e59T + 1.75e119T^{2} \) |
| 47 | \( 1 - 2.87e60T + 1.15e122T^{2} \) |
| 53 | \( 1 - 1.26e63T + 7.44e125T^{2} \) |
| 59 | \( 1 + 1.90e64T + 1.87e129T^{2} \) |
| 61 | \( 1 + 1.99e65T + 2.13e130T^{2} \) |
| 67 | \( 1 - 1.27e66T + 2.01e133T^{2} \) |
| 71 | \( 1 + 5.67e67T + 1.38e135T^{2} \) |
| 73 | \( 1 + 3.49e67T + 1.05e136T^{2} \) |
| 79 | \( 1 + 3.25e69T + 3.36e138T^{2} \) |
| 83 | \( 1 - 2.12e70T + 1.23e140T^{2} \) |
| 89 | \( 1 - 5.59e70T + 2.02e142T^{2} \) |
| 97 | \( 1 + 1.66e72T + 1.08e145T^{2} \) |
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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
−13.13836352711591502182145799578, −11.30193706570880635085387165995, −10.62888852381961838494519931772, −8.977723660216591352630663012940, −7.47141372178201152864596624219, −5.82454343625433112359562989855, −5.02155636073421785653005595409, −2.47442393605397948668074291459, −1.32949285260893221464985841136, 0,
1.32949285260893221464985841136, 2.47442393605397948668074291459, 5.02155636073421785653005595409, 5.82454343625433112359562989855, 7.47141372178201152864596624219, 8.977723660216591352630663012940, 10.62888852381961838494519931772, 11.30193706570880635085387165995, 13.13836352711591502182145799578
