| L(s) = 1 | + 0.966·2-s + 1.18·3-s − 1.06·4-s + 3.99·5-s + 1.14·6-s + 0.596·7-s − 2.96·8-s − 1.60·9-s + 3.86·10-s + 5.38·11-s − 1.25·12-s − 0.870·13-s + 0.576·14-s + 4.71·15-s − 0.731·16-s − 1.89·17-s − 1.55·18-s + 2.15·19-s − 4.25·20-s + 0.704·21-s + 5.20·22-s + 2.92·23-s − 3.50·24-s + 10.9·25-s − 0.841·26-s − 5.43·27-s − 0.635·28-s + ⋯ |
| L(s) = 1 | + 0.683·2-s + 0.682·3-s − 0.532·4-s + 1.78·5-s + 0.466·6-s + 0.225·7-s − 1.04·8-s − 0.534·9-s + 1.22·10-s + 1.62·11-s − 0.363·12-s − 0.241·13-s + 0.154·14-s + 1.21·15-s − 0.182·16-s − 0.460·17-s − 0.365·18-s + 0.494·19-s − 0.952·20-s + 0.153·21-s + 1.11·22-s + 0.609·23-s − 0.714·24-s + 2.19·25-s − 0.164·26-s − 1.04·27-s − 0.120·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1369 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1369 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.486415933\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.486415933\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 \) |
| good | 2 | \( 1 - 0.966T + 2T^{2} \) |
| 3 | \( 1 - 1.18T + 3T^{2} \) |
| 5 | \( 1 - 3.99T + 5T^{2} \) |
| 7 | \( 1 - 0.596T + 7T^{2} \) |
| 11 | \( 1 - 5.38T + 11T^{2} \) |
| 13 | \( 1 + 0.870T + 13T^{2} \) |
| 17 | \( 1 + 1.89T + 17T^{2} \) |
| 19 | \( 1 - 2.15T + 19T^{2} \) |
| 23 | \( 1 - 2.92T + 23T^{2} \) |
| 29 | \( 1 - 4.57T + 29T^{2} \) |
| 31 | \( 1 + 0.253T + 31T^{2} \) |
| 41 | \( 1 - 6.55T + 41T^{2} \) |
| 43 | \( 1 + 7.71T + 43T^{2} \) |
| 47 | \( 1 + 5.38T + 47T^{2} \) |
| 53 | \( 1 + 7.24T + 53T^{2} \) |
| 59 | \( 1 - 3.93T + 59T^{2} \) |
| 61 | \( 1 - 11.6T + 61T^{2} \) |
| 67 | \( 1 - 4.11T + 67T^{2} \) |
| 71 | \( 1 + 9.61T + 71T^{2} \) |
| 73 | \( 1 + 12.0T + 73T^{2} \) |
| 79 | \( 1 - 0.740T + 79T^{2} \) |
| 83 | \( 1 + 5.72T + 83T^{2} \) |
| 89 | \( 1 - 11.0T + 89T^{2} \) |
| 97 | \( 1 - 11.7T + 97T^{2} \) |
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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
−9.424098100524731312272323273842, −8.993688701807049700693612576287, −8.306333205447988517819314417413, −6.78716036176239784741245112462, −6.17750664264119024894741773525, −5.37385718430704317345306073520, −4.57963286300168097051918271495, −3.42399947273380855767129969987, −2.56735390115225979537868776219, −1.39765026989014382833316645918,
1.39765026989014382833316645918, 2.56735390115225979537868776219, 3.42399947273380855767129969987, 4.57963286300168097051918271495, 5.37385718430704317345306073520, 6.17750664264119024894741773525, 6.78716036176239784741245112462, 8.306333205447988517819314417413, 8.993688701807049700693612576287, 9.424098100524731312272323273842
