| L(s) = 1 | − 2-s + 3.04·3-s + 4-s − 1.82·5-s − 3.04·6-s − 0.477·7-s − 8-s + 6.24·9-s + 1.82·10-s − 2.35·11-s + 3.04·12-s + 0.195·13-s + 0.477·14-s − 5.56·15-s + 16-s − 0.533·17-s − 6.24·18-s − 0.859·19-s − 1.82·20-s − 1.45·21-s + 2.35·22-s − 5.65·23-s − 3.04·24-s − 1.65·25-s − 0.195·26-s + 9.87·27-s − 0.477·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.75·3-s + 0.5·4-s − 0.818·5-s − 1.24·6-s − 0.180·7-s − 0.353·8-s + 2.08·9-s + 0.578·10-s − 0.709·11-s + 0.877·12-s + 0.0542·13-s + 0.127·14-s − 1.43·15-s + 0.250·16-s − 0.129·17-s − 1.47·18-s − 0.197·19-s − 0.409·20-s − 0.316·21-s + 0.501·22-s − 1.17·23-s − 0.620·24-s − 0.330·25-s − 0.0383·26-s + 1.89·27-s − 0.0902·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 2011 | \( 1 + T \) |
| good | 3 | \( 1 - 3.04T + 3T^{2} \) |
| 5 | \( 1 + 1.82T + 5T^{2} \) |
| 7 | \( 1 + 0.477T + 7T^{2} \) |
| 11 | \( 1 + 2.35T + 11T^{2} \) |
| 13 | \( 1 - 0.195T + 13T^{2} \) |
| 17 | \( 1 + 0.533T + 17T^{2} \) |
| 19 | \( 1 + 0.859T + 19T^{2} \) |
| 23 | \( 1 + 5.65T + 23T^{2} \) |
| 29 | \( 1 - 4.86T + 29T^{2} \) |
| 31 | \( 1 + 7.78T + 31T^{2} \) |
| 37 | \( 1 + 2.89T + 37T^{2} \) |
| 41 | \( 1 + 5.00T + 41T^{2} \) |
| 43 | \( 1 + 7.29T + 43T^{2} \) |
| 47 | \( 1 + 0.152T + 47T^{2} \) |
| 53 | \( 1 - 8.94T + 53T^{2} \) |
| 59 | \( 1 - 6.95T + 59T^{2} \) |
| 61 | \( 1 + 5.59T + 61T^{2} \) |
| 67 | \( 1 - 2.97T + 67T^{2} \) |
| 71 | \( 1 + 10.8T + 71T^{2} \) |
| 73 | \( 1 + 6.96T + 73T^{2} \) |
| 79 | \( 1 + 10.1T + 79T^{2} \) |
| 83 | \( 1 - 3.82T + 83T^{2} \) |
| 89 | \( 1 - 3.33T + 89T^{2} \) |
| 97 | \( 1 - 15.0T + 97T^{2} \) |
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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
−8.341523141057279423260733084786, −7.52375435384130134055717011797, −7.15768695903309132841227025930, −6.10254182091852633613673775177, −4.85344259159312499611794509758, −3.85355655455852672774591602980, −3.34609424925429932945376739076, −2.43437946318621867981811459245, −1.66972234409657086408463005759, 0,
1.66972234409657086408463005759, 2.43437946318621867981811459245, 3.34609424925429932945376739076, 3.85355655455852672774591602980, 4.85344259159312499611794509758, 6.10254182091852633613673775177, 7.15768695903309132841227025930, 7.52375435384130134055717011797, 8.341523141057279423260733084786
