| L(s) = 1 | − 1.27·2-s + 1.65·3-s − 0.377·4-s − 2.10·6-s + 3.65·7-s + 3.02·8-s − 0.273·9-s + 2.65·11-s − 0.622·12-s − 6.13·13-s − 4.65·14-s − 3.10·16-s + 2.34·17-s + 0.348·18-s + 6.02·21-s − 3.37·22-s + 5.48·23-s + 5·24-s + 7.81·26-s − 5.40·27-s − 1.37·28-s − 0.651·29-s + 6.67·31-s − 2.10·32-s + 4.37·33-s − 2.99·34-s + 0.103·36-s + ⋯ |
| L(s) = 1 | − 0.900·2-s + 0.953·3-s − 0.188·4-s − 0.858·6-s + 1.37·7-s + 1.07·8-s − 0.0912·9-s + 0.799·11-s − 0.179·12-s − 1.70·13-s − 1.24·14-s − 0.775·16-s + 0.569·17-s + 0.0822·18-s + 1.31·21-s − 0.720·22-s + 1.14·23-s + 1.02·24-s + 1.53·26-s − 1.04·27-s − 0.260·28-s − 0.120·29-s + 1.19·31-s − 0.371·32-s + 0.761·33-s − 0.513·34-s + 0.0172·36-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
| \(L(1)\) |
\(\approx\) |
\(1.903353457\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.903353457\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.27T + 2T^{2} \) |
| 3 | \( 1 - 1.65T + 3T^{2} \) |
| 7 | \( 1 - 3.65T + 7T^{2} \) |
| 11 | \( 1 - 2.65T + 11T^{2} \) |
| 13 | \( 1 + 6.13T + 13T^{2} \) |
| 17 | \( 1 - 2.34T + 17T^{2} \) |
| 23 | \( 1 - 5.48T + 23T^{2} \) |
| 29 | \( 1 + 0.651T + 29T^{2} \) |
| 31 | \( 1 - 6.67T + 31T^{2} \) |
| 37 | \( 1 - 8.70T + 37T^{2} \) |
| 41 | \( 1 + 1.93T + 41T^{2} \) |
| 43 | \( 1 + 2.65T + 43T^{2} \) |
| 47 | \( 1 - 3.71T + 47T^{2} \) |
| 53 | \( 1 + 13.7T + 53T^{2} \) |
| 59 | \( 1 - 7.84T + 59T^{2} \) |
| 61 | \( 1 + 1.92T + 61T^{2} \) |
| 67 | \( 1 - 4.44T + 67T^{2} \) |
| 71 | \( 1 + 3.54T + 71T^{2} \) |
| 73 | \( 1 + 2.48T + 73T^{2} \) |
| 79 | \( 1 - 15.1T + 79T^{2} \) |
| 83 | \( 1 + 14.7T + 83T^{2} \) |
| 89 | \( 1 - 5.06T + 89T^{2} \) |
| 97 | \( 1 + 3.22T + 97T^{2} \) |
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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
−7.83529402654415985654975562389, −7.52657927654992508840082344384, −6.71218232747320705937821126001, −5.52651061405222744608190411249, −4.75545163118876399013881011901, −4.40857586616869880911949911513, −3.29079235925564098254827503198, −2.42908943285583003003253071415, −1.67277236987825611638625001988, −0.76640530447595847813800203283,
0.76640530447595847813800203283, 1.67277236987825611638625001988, 2.42908943285583003003253071415, 3.29079235925564098254827503198, 4.40857586616869880911949911513, 4.75545163118876399013881011901, 5.52651061405222744608190411249, 6.71218232747320705937821126001, 7.52657927654992508840082344384, 7.83529402654415985654975562389
