| L(s) = 1 | − 1.00·2-s + 2.95·3-s − 0.996·4-s − 2.95·6-s − 1.49·7-s + 3.00·8-s + 5.70·9-s + 0.117·11-s − 2.94·12-s − 5.56·13-s + 1.49·14-s − 1.01·16-s − 3.01·17-s − 5.71·18-s + 4.81·19-s − 4.40·21-s − 0.117·22-s − 8.22·23-s + 8.85·24-s + 5.57·26-s + 7.98·27-s + 1.48·28-s + 6.36·29-s − 0.882·31-s − 4.98·32-s + 0.347·33-s + 3.02·34-s + ⋯ |
| L(s) = 1 | − 0.708·2-s + 1.70·3-s − 0.498·4-s − 1.20·6-s − 0.564·7-s + 1.06·8-s + 1.90·9-s + 0.0354·11-s − 0.848·12-s − 1.54·13-s + 0.399·14-s − 0.253·16-s − 0.731·17-s − 1.34·18-s + 1.10·19-s − 0.961·21-s − 0.0251·22-s − 1.71·23-s + 1.80·24-s + 1.09·26-s + 1.53·27-s + 0.281·28-s + 1.18·29-s − 0.158·31-s − 0.881·32-s + 0.0604·33-s + 0.518·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 241 | \( 1 - T \) |
| good | 2 | \( 1 + 1.00T + 2T^{2} \) |
| 3 | \( 1 - 2.95T + 3T^{2} \) |
| 7 | \( 1 + 1.49T + 7T^{2} \) |
| 11 | \( 1 - 0.117T + 11T^{2} \) |
| 13 | \( 1 + 5.56T + 13T^{2} \) |
| 17 | \( 1 + 3.01T + 17T^{2} \) |
| 19 | \( 1 - 4.81T + 19T^{2} \) |
| 23 | \( 1 + 8.22T + 23T^{2} \) |
| 29 | \( 1 - 6.36T + 29T^{2} \) |
| 31 | \( 1 + 0.882T + 31T^{2} \) |
| 37 | \( 1 - 6.53T + 37T^{2} \) |
| 41 | \( 1 + 7.03T + 41T^{2} \) |
| 43 | \( 1 - 12.9T + 43T^{2} \) |
| 47 | \( 1 - 6.49T + 47T^{2} \) |
| 53 | \( 1 + 9.80T + 53T^{2} \) |
| 59 | \( 1 - 10.1T + 59T^{2} \) |
| 61 | \( 1 + 2.66T + 61T^{2} \) |
| 67 | \( 1 + 12.4T + 67T^{2} \) |
| 71 | \( 1 + 0.718T + 71T^{2} \) |
| 73 | \( 1 + 15.3T + 73T^{2} \) |
| 79 | \( 1 - 9.25T + 79T^{2} \) |
| 83 | \( 1 + 2.92T + 83T^{2} \) |
| 89 | \( 1 + 16.1T + 89T^{2} \) |
| 97 | \( 1 - 0.213T + 97T^{2} \) |
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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
−7.87979590269969459039241450593, −7.42497731573294451031979510657, −6.69471749481764448965357214910, −5.50923935169140516169880866309, −4.43742479109480782034315113384, −4.08697437659245532210525741614, −2.96773057867110805997076385741, −2.43835959992985072152465653850, −1.41646578108251024252817344351, 0,
1.41646578108251024252817344351, 2.43835959992985072152465653850, 2.96773057867110805997076385741, 4.08697437659245532210525741614, 4.43742479109480782034315113384, 5.50923935169140516169880866309, 6.69471749481764448965357214910, 7.42497731573294451031979510657, 7.87979590269969459039241450593
