L(s) = 1 | + 2-s + 0.572·3-s + 4-s + 2.21·5-s + 0.572·6-s + 7-s + 8-s − 2.67·9-s + 2.21·10-s + 6.09·11-s + 0.572·12-s + 6.15·13-s + 14-s + 1.26·15-s + 16-s − 1.73·17-s − 2.67·18-s + 2.21·20-s + 0.572·21-s + 6.09·22-s − 3.38·23-s + 0.572·24-s − 0.0738·25-s + 6.15·26-s − 3.24·27-s + 28-s − 2.18·29-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.330·3-s + 0.5·4-s + 0.992·5-s + 0.233·6-s + 0.377·7-s + 0.353·8-s − 0.890·9-s + 0.701·10-s + 1.83·11-s + 0.165·12-s + 1.70·13-s + 0.267·14-s + 0.327·15-s + 0.250·16-s − 0.421·17-s − 0.629·18-s + 0.496·20-s + 0.124·21-s + 1.29·22-s − 0.706·23-s + 0.116·24-s − 0.0147·25-s + 1.20·26-s − 0.624·27-s + 0.188·28-s − 0.406·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.112329139\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.112329139\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 \) |
good | 3 | \( 1 - 0.572T + 3T^{2} \) |
| 5 | \( 1 - 2.21T + 5T^{2} \) |
| 11 | \( 1 - 6.09T + 11T^{2} \) |
| 13 | \( 1 - 6.15T + 13T^{2} \) |
| 17 | \( 1 + 1.73T + 17T^{2} \) |
| 23 | \( 1 + 3.38T + 23T^{2} \) |
| 29 | \( 1 + 2.18T + 29T^{2} \) |
| 31 | \( 1 + 0.977T + 31T^{2} \) |
| 37 | \( 1 - 11.9T + 37T^{2} \) |
| 41 | \( 1 + 8.25T + 41T^{2} \) |
| 43 | \( 1 - 3.28T + 43T^{2} \) |
| 47 | \( 1 - 9.95T + 47T^{2} \) |
| 53 | \( 1 - 3.81T + 53T^{2} \) |
| 59 | \( 1 + 3.92T + 59T^{2} \) |
| 61 | \( 1 - 1.40T + 61T^{2} \) |
| 67 | \( 1 + 1.83T + 67T^{2} \) |
| 71 | \( 1 + 16.2T + 71T^{2} \) |
| 73 | \( 1 + 8.95T + 73T^{2} \) |
| 79 | \( 1 + 15.3T + 79T^{2} \) |
| 83 | \( 1 - 11.7T + 83T^{2} \) |
| 89 | \( 1 + 13.2T + 89T^{2} \) |
| 97 | \( 1 - 11.4T + 97T^{2} \) |
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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
−8.394414666773749713496635612334, −7.41607222055054132209736764795, −6.42893454709344829663938219277, −6.01178317651192624770634035842, −5.60938452867813361928221854726, −4.28720299233461093025375558734, −3.87764748879034373269785828908, −2.91587318819192095478262282873, −1.93200820130251663684501075768, −1.23961422095054891612649500910,
1.23961422095054891612649500910, 1.93200820130251663684501075768, 2.91587318819192095478262282873, 3.87764748879034373269785828908, 4.28720299233461093025375558734, 5.60938452867813361928221854726, 6.01178317651192624770634035842, 6.42893454709344829663938219277, 7.41607222055054132209736764795, 8.394414666773749713496635612334