| L(s) = 1 | − 2-s − 0.0638·3-s + 4-s + 5-s + 0.0638·6-s + 2.62·7-s − 8-s − 2.99·9-s − 10-s − 11-s − 0.0638·12-s − 1.25·13-s − 2.62·14-s − 0.0638·15-s + 16-s + 3.93·17-s + 2.99·18-s − 2.36·19-s + 20-s − 0.167·21-s + 22-s − 3.68·23-s + 0.0638·24-s + 25-s + 1.25·26-s + 0.382·27-s + 2.62·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.0368·3-s + 0.5·4-s + 0.447·5-s + 0.0260·6-s + 0.991·7-s − 0.353·8-s − 0.998·9-s − 0.316·10-s − 0.301·11-s − 0.0184·12-s − 0.349·13-s − 0.701·14-s − 0.0164·15-s + 0.250·16-s + 0.954·17-s + 0.706·18-s − 0.543·19-s + 0.223·20-s − 0.0365·21-s + 0.213·22-s − 0.768·23-s + 0.0130·24-s + 0.200·25-s + 0.246·26-s + 0.0736·27-s + 0.495·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.488051919\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.488051919\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 73 | \( 1 + T \) |
| good | 3 | \( 1 + 0.0638T + 3T^{2} \) |
| 7 | \( 1 - 2.62T + 7T^{2} \) |
| 13 | \( 1 + 1.25T + 13T^{2} \) |
| 17 | \( 1 - 3.93T + 17T^{2} \) |
| 19 | \( 1 + 2.36T + 19T^{2} \) |
| 23 | \( 1 + 3.68T + 23T^{2} \) |
| 29 | \( 1 - 5.88T + 29T^{2} \) |
| 31 | \( 1 - 1.48T + 31T^{2} \) |
| 37 | \( 1 - 4.65T + 37T^{2} \) |
| 41 | \( 1 - 11.3T + 41T^{2} \) |
| 43 | \( 1 + 9.37T + 43T^{2} \) |
| 47 | \( 1 + 2.71T + 47T^{2} \) |
| 53 | \( 1 - 10.5T + 53T^{2} \) |
| 59 | \( 1 + 9.86T + 59T^{2} \) |
| 61 | \( 1 + 9.36T + 61T^{2} \) |
| 67 | \( 1 + 4.79T + 67T^{2} \) |
| 71 | \( 1 - 9.41T + 71T^{2} \) |
| 79 | \( 1 + 0.446T + 79T^{2} \) |
| 83 | \( 1 + 16.2T + 83T^{2} \) |
| 89 | \( 1 - 8.94T + 89T^{2} \) |
| 97 | \( 1 - 14.9T + 97T^{2} \) |
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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
−7.908604956385110837087534595116, −7.43807846231024145819896598376, −6.31322218520926312308443389401, −5.93115294542203944664211970725, −5.10406038582852815326167995084, −4.45264573886223051233021418726, −3.23040168176619735857812745599, −2.50349732849894342105645829394, −1.73324164043097765476949434547, −0.67418638414870216570353765786,
0.67418638414870216570353765786, 1.73324164043097765476949434547, 2.50349732849894342105645829394, 3.23040168176619735857812745599, 4.45264573886223051233021418726, 5.10406038582852815326167995084, 5.93115294542203944664211970725, 6.31322218520926312308443389401, 7.43807846231024145819896598376, 7.908604956385110837087534595116
