| L(s) = 1 | + 2-s + 1.61·3-s + 4-s + 5-s + 1.61·6-s − 0.618·7-s + 8-s − 0.381·9-s + 10-s − 2.85·11-s + 1.61·12-s − 7.09·13-s − 0.618·14-s + 1.61·15-s + 16-s + 6.09·17-s − 0.381·18-s + 1.85·19-s + 20-s − 1.00·21-s − 2.85·22-s + 23-s + 1.61·24-s + 25-s − 7.09·26-s − 5.47·27-s − 0.618·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.934·3-s + 0.5·4-s + 0.447·5-s + 0.660·6-s − 0.233·7-s + 0.353·8-s − 0.127·9-s + 0.316·10-s − 0.860·11-s + 0.467·12-s − 1.96·13-s − 0.165·14-s + 0.417·15-s + 0.250·16-s + 1.47·17-s − 0.0900·18-s + 0.425·19-s + 0.223·20-s − 0.218·21-s − 0.608·22-s + 0.208·23-s + 0.330·24-s + 0.200·25-s − 1.39·26-s − 1.05·27-s − 0.116·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.253074220\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.253074220\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 - 1.61T + 3T^{2} \) |
| 7 | \( 1 + 0.618T + 7T^{2} \) |
| 11 | \( 1 + 2.85T + 11T^{2} \) |
| 13 | \( 1 + 7.09T + 13T^{2} \) |
| 17 | \( 1 - 6.09T + 17T^{2} \) |
| 19 | \( 1 - 1.85T + 19T^{2} \) |
| 29 | \( 1 + 9.23T + 29T^{2} \) |
| 31 | \( 1 - 9.09T + 31T^{2} \) |
| 37 | \( 1 - 6.47T + 37T^{2} \) |
| 41 | \( 1 - 3.32T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 3.70T + 47T^{2} \) |
| 53 | \( 1 - 0.472T + 53T^{2} \) |
| 59 | \( 1 - 1.70T + 59T^{2} \) |
| 61 | \( 1 + 9.32T + 61T^{2} \) |
| 67 | \( 1 - 14.4T + 67T^{2} \) |
| 71 | \( 1 + 4.09T + 71T^{2} \) |
| 73 | \( 1 - 3.23T + 73T^{2} \) |
| 79 | \( 1 - 1.52T + 79T^{2} \) |
| 83 | \( 1 + 6.94T + 83T^{2} \) |
| 89 | \( 1 + 10.4T + 89T^{2} \) |
| 97 | \( 1 - 12.3T + 97T^{2} \) |
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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
−12.47883434393512256328344394257, −11.44686251142789410021465049921, −10.01325729367604094589154766938, −9.542030720090398116833551036626, −7.996795640306081590736648146812, −7.37284920887802757557061253147, −5.81865497788505764214963526118, −4.88440935172606354314966556919, −3.21835418156143581887086605491, −2.38964411962337051053951412739,
2.38964411962337051053951412739, 3.21835418156143581887086605491, 4.88440935172606354314966556919, 5.81865497788505764214963526118, 7.37284920887802757557061253147, 7.996795640306081590736648146812, 9.542030720090398116833551036626, 10.01325729367604094589154766938, 11.44686251142789410021465049921, 12.47883434393512256328344394257
