| L(s) = 1 | + 0.571·2-s − 3-s − 1.67·4-s − 5-s − 0.571·6-s + 2.42·7-s − 2.10·8-s + 9-s − 0.571·10-s + 1.14·11-s + 1.67·12-s + 1.57·13-s + 1.38·14-s + 15-s + 2.14·16-s + 4.67·17-s + 0.571·18-s + 5.34·19-s + 1.67·20-s − 2.42·21-s + 0.654·22-s − 1.81·23-s + 2.10·24-s + 25-s + 0.899·26-s − 27-s − 4.06·28-s + ⋯ |
| L(s) = 1 | + 0.404·2-s − 0.577·3-s − 0.836·4-s − 0.447·5-s − 0.233·6-s + 0.917·7-s − 0.742·8-s + 0.333·9-s − 0.180·10-s + 0.344·11-s + 0.482·12-s + 0.435·13-s + 0.371·14-s + 0.258·15-s + 0.535·16-s + 1.13·17-s + 0.134·18-s + 1.22·19-s + 0.374·20-s − 0.529·21-s + 0.139·22-s − 0.378·23-s + 0.428·24-s + 0.200·25-s + 0.176·26-s − 0.192·27-s − 0.767·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.234145367\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.234145367\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 - 0.571T + 2T^{2} \) |
| 7 | \( 1 - 2.42T + 7T^{2} \) |
| 11 | \( 1 - 1.14T + 11T^{2} \) |
| 13 | \( 1 - 1.57T + 13T^{2} \) |
| 17 | \( 1 - 4.67T + 17T^{2} \) |
| 19 | \( 1 - 5.34T + 19T^{2} \) |
| 23 | \( 1 + 1.81T + 23T^{2} \) |
| 29 | \( 1 - 1.95T + 29T^{2} \) |
| 37 | \( 1 - 3.57T + 37T^{2} \) |
| 41 | \( 1 + 3.14T + 41T^{2} \) |
| 43 | \( 1 - 2.85T + 43T^{2} \) |
| 47 | \( 1 - 7.16T + 47T^{2} \) |
| 53 | \( 1 + 3.81T + 53T^{2} \) |
| 59 | \( 1 + 6.24T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 0.917T + 67T^{2} \) |
| 71 | \( 1 + 6.44T + 71T^{2} \) |
| 73 | \( 1 - 6.71T + 73T^{2} \) |
| 79 | \( 1 - 9.52T + 79T^{2} \) |
| 83 | \( 1 + 3.16T + 83T^{2} \) |
| 89 | \( 1 + 15.2T + 89T^{2} \) |
| 97 | \( 1 - 10.7T + 97T^{2} \) |
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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
−11.26728516744481961819582277088, −10.18602271935891487914742397084, −9.282367605206124264519674661134, −8.234748983977437497255673065667, −7.50166902945230036054585222746, −6.06909831289549431663979977821, −5.22346080809494638330064455958, −4.38555451146213858449669391566, −3.34757205980967648124465139881, −1.09640622374071572592253444859,
1.09640622374071572592253444859, 3.34757205980967648124465139881, 4.38555451146213858449669391566, 5.22346080809494638330064455958, 6.06909831289549431663979977821, 7.50166902945230036054585222746, 8.234748983977437497255673065667, 9.282367605206124264519674661134, 10.18602271935891487914742397084, 11.26728516744481961819582277088
