| L(s) = 1 | + 2-s − 3.23·3-s + 4-s − 3.23·6-s + 7-s + 8-s + 7.47·9-s + 5.23·11-s − 3.23·12-s + 2.47·13-s + 14-s + 16-s − 17-s + 7.47·18-s − 8.47·19-s − 3.23·21-s + 5.23·22-s − 8·23-s − 3.23·24-s + 2.47·26-s − 14.4·27-s + 28-s − 5.70·29-s − 1.52·31-s + 32-s − 16.9·33-s − 34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.86·3-s + 0.5·4-s − 1.32·6-s + 0.377·7-s + 0.353·8-s + 2.49·9-s + 1.57·11-s − 0.934·12-s + 0.685·13-s + 0.267·14-s + 0.250·16-s − 0.242·17-s + 1.76·18-s − 1.94·19-s − 0.706·21-s + 1.11·22-s − 1.66·23-s − 0.660·24-s + 0.484·26-s − 2.78·27-s + 0.188·28-s − 1.05·29-s − 0.274·31-s + 0.176·32-s − 2.94·33-s − 0.171·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 + 3.23T + 3T^{2} \) |
| 11 | \( 1 - 5.23T + 11T^{2} \) |
| 13 | \( 1 - 2.47T + 13T^{2} \) |
| 19 | \( 1 + 8.47T + 19T^{2} \) |
| 23 | \( 1 + 8T + 23T^{2} \) |
| 29 | \( 1 + 5.70T + 29T^{2} \) |
| 31 | \( 1 + 1.52T + 31T^{2} \) |
| 37 | \( 1 - 3.23T + 37T^{2} \) |
| 41 | \( 1 + 3.52T + 41T^{2} \) |
| 43 | \( 1 + 2.47T + 43T^{2} \) |
| 47 | \( 1 - 2.47T + 47T^{2} \) |
| 53 | \( 1 - 4.47T + 53T^{2} \) |
| 59 | \( 1 + 6T + 59T^{2} \) |
| 61 | \( 1 - 1.23T + 61T^{2} \) |
| 67 | \( 1 - 1.52T + 67T^{2} \) |
| 71 | \( 1 + 2.47T + 71T^{2} \) |
| 73 | \( 1 + 13.4T + 73T^{2} \) |
| 79 | \( 1 + 10.4T + 79T^{2} \) |
| 83 | \( 1 + 2T + 83T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 - 8.47T + 97T^{2} \) |
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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
−7.33494719551770040165350950377, −6.67725483973559868612194058936, −6.04019892997750444157409167749, −5.89350982574938464282746485527, −4.80294342093957061906822588346, −4.12232741272564039087198748468, −3.85964005600496296816654739047, −2.00346828783341979017689215199, −1.36826581247304434124664115743, 0,
1.36826581247304434124664115743, 2.00346828783341979017689215199, 3.85964005600496296816654739047, 4.12232741272564039087198748468, 4.80294342093957061906822588346, 5.89350982574938464282746485527, 6.04019892997750444157409167749, 6.67725483973559868612194058936, 7.33494719551770040165350950377
