| L(s) = 1 | − 2.23·2-s + 3.00·4-s + 5-s + 2·7-s − 2.23·8-s − 2.23·10-s + 2·11-s + 2·13-s − 4.47·14-s − 0.999·16-s + 4.47·17-s + 2·19-s + 3.00·20-s − 4.47·22-s + 2·23-s + 25-s − 4.47·26-s + 6.00·28-s + 29-s − 2·31-s + 6.70·32-s − 10.0·34-s + 2·35-s − 0.472·37-s − 4.47·38-s − 2.23·40-s − 2·41-s + ⋯ |
| L(s) = 1 | − 1.58·2-s + 1.50·4-s + 0.447·5-s + 0.755·7-s − 0.790·8-s − 0.707·10-s + 0.603·11-s + 0.554·13-s − 1.19·14-s − 0.249·16-s + 1.08·17-s + 0.458·19-s + 0.670·20-s − 0.953·22-s + 0.417·23-s + 0.200·25-s − 0.877·26-s + 1.13·28-s + 0.185·29-s − 0.359·31-s + 1.18·32-s − 1.71·34-s + 0.338·35-s − 0.0776·37-s − 0.725·38-s − 0.353·40-s − 0.312·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.029920863\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.029920863\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 + 2.23T + 2T^{2} \) |
| 7 | \( 1 - 2T + 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 - 4.47T + 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 - 2T + 23T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 + 0.472T + 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 - 4.94T + 47T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 + 8T + 59T^{2} \) |
| 61 | \( 1 + 10.9T + 61T^{2} \) |
| 67 | \( 1 + 14.9T + 67T^{2} \) |
| 71 | \( 1 + 4T + 71T^{2} \) |
| 73 | \( 1 - 12.4T + 73T^{2} \) |
| 79 | \( 1 - 14.9T + 79T^{2} \) |
| 83 | \( 1 + 2.94T + 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 - 9.41T + 97T^{2} \) |
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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
−9.385445675641213343479916247367, −9.045744038333210674578775402860, −8.072025987745776821115894533478, −7.55837890506713340163497813183, −6.60996500575213862476131281066, −5.70256981899790712092527893901, −4.59323310609596032035838359014, −3.20019221173085395800896109132, −1.81905966326364169270266977095, −1.03856991848935171518904010290,
1.03856991848935171518904010290, 1.81905966326364169270266977095, 3.20019221173085395800896109132, 4.59323310609596032035838359014, 5.70256981899790712092527893901, 6.60996500575213862476131281066, 7.55837890506713340163497813183, 8.072025987745776821115894533478, 9.045744038333210674578775402860, 9.385445675641213343479916247367
