| L(s) = 1 | − 0.862·2-s − 3.07·3-s − 1.25·4-s + 2.65·6-s − 0.566·7-s + 2.80·8-s + 6.48·9-s − 1.91·11-s + 3.86·12-s − 0.194·13-s + 0.488·14-s + 0.0877·16-s − 5.29·17-s − 5.59·18-s + 1.74·21-s + 1.64·22-s − 3.37·23-s − 8.64·24-s + 0.167·26-s − 10.7·27-s + 0.711·28-s − 8.73·29-s + 5.65·31-s − 5.69·32-s + 5.88·33-s + 4.56·34-s − 8.13·36-s + ⋯ |
| L(s) = 1 | − 0.610·2-s − 1.77·3-s − 0.627·4-s + 1.08·6-s − 0.214·7-s + 0.993·8-s + 2.16·9-s − 0.576·11-s + 1.11·12-s − 0.0539·13-s + 0.130·14-s + 0.0219·16-s − 1.28·17-s − 1.31·18-s + 0.380·21-s + 0.351·22-s − 0.703·23-s − 1.76·24-s + 0.0329·26-s − 2.06·27-s + 0.134·28-s − 1.62·29-s + 1.01·31-s − 1.00·32-s + 1.02·33-s + 0.783·34-s − 1.35·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1630222556\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1630222556\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + 0.862T + 2T^{2} \) |
| 3 | \( 1 + 3.07T + 3T^{2} \) |
| 7 | \( 1 + 0.566T + 7T^{2} \) |
| 11 | \( 1 + 1.91T + 11T^{2} \) |
| 13 | \( 1 + 0.194T + 13T^{2} \) |
| 17 | \( 1 + 5.29T + 17T^{2} \) |
| 23 | \( 1 + 3.37T + 23T^{2} \) |
| 29 | \( 1 + 8.73T + 29T^{2} \) |
| 31 | \( 1 - 5.65T + 31T^{2} \) |
| 37 | \( 1 + 0.955T + 37T^{2} \) |
| 41 | \( 1 - 10.0T + 41T^{2} \) |
| 43 | \( 1 - 4.93T + 43T^{2} \) |
| 47 | \( 1 + 8.83T + 47T^{2} \) |
| 53 | \( 1 - 8.20T + 53T^{2} \) |
| 59 | \( 1 - 3.71T + 59T^{2} \) |
| 61 | \( 1 + 3.51T + 61T^{2} \) |
| 67 | \( 1 + 4.04T + 67T^{2} \) |
| 71 | \( 1 - 5.19T + 71T^{2} \) |
| 73 | \( 1 + 8.60T + 73T^{2} \) |
| 79 | \( 1 - 6.62T + 79T^{2} \) |
| 83 | \( 1 - 4.51T + 83T^{2} \) |
| 89 | \( 1 - 3.37T + 89T^{2} \) |
| 97 | \( 1 + 15.1T + 97T^{2} \) |
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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
−7.65755547290975161895643336189, −7.04077754387783607456457753613, −6.28247703189959636329043615807, −5.70926578538377851021089367234, −5.01416305197642682972898681656, −4.43836955361695719732384198504, −3.82153843271121289665511385347, −2.32565355573114277074185361834, −1.29244133191854249630279506874, −0.26240750598271101047829499935,
0.26240750598271101047829499935, 1.29244133191854249630279506874, 2.32565355573114277074185361834, 3.82153843271121289665511385347, 4.43836955361695719732384198504, 5.01416305197642682972898681656, 5.70926578538377851021089367234, 6.28247703189959636329043615807, 7.04077754387783607456457753613, 7.65755547290975161895643336189
