| L(s) = 1 | − 2-s + 2.80·3-s + 4-s − 5-s − 2.80·6-s − 2.29·7-s − 8-s + 4.87·9-s + 10-s − 0.201·11-s + 2.80·12-s − 13-s + 2.29·14-s − 2.80·15-s + 16-s − 3.76·17-s − 4.87·18-s − 2.78·19-s − 20-s − 6.42·21-s + 0.201·22-s + 7.54·23-s − 2.80·24-s + 25-s + 26-s + 5.25·27-s − 2.29·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.61·3-s + 0.5·4-s − 0.447·5-s − 1.14·6-s − 0.865·7-s − 0.353·8-s + 1.62·9-s + 0.316·10-s − 0.0608·11-s + 0.809·12-s − 0.277·13-s + 0.612·14-s − 0.724·15-s + 0.250·16-s − 0.914·17-s − 1.14·18-s − 0.639·19-s − 0.223·20-s − 1.40·21-s + 0.0430·22-s + 1.57·23-s − 0.572·24-s + 0.200·25-s + 0.196·26-s + 1.01·27-s − 0.432·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 31 | \( 1 + T \) |
| good | 3 | \( 1 - 2.80T + 3T^{2} \) |
| 7 | \( 1 + 2.29T + 7T^{2} \) |
| 11 | \( 1 + 0.201T + 11T^{2} \) |
| 17 | \( 1 + 3.76T + 17T^{2} \) |
| 19 | \( 1 + 2.78T + 19T^{2} \) |
| 23 | \( 1 - 7.54T + 23T^{2} \) |
| 29 | \( 1 + 2.95T + 29T^{2} \) |
| 37 | \( 1 + 8.32T + 37T^{2} \) |
| 41 | \( 1 - 4.88T + 41T^{2} \) |
| 43 | \( 1 + 4.86T + 43T^{2} \) |
| 47 | \( 1 + 0.508T + 47T^{2} \) |
| 53 | \( 1 - 5.48T + 53T^{2} \) |
| 59 | \( 1 + 9.36T + 59T^{2} \) |
| 61 | \( 1 + 2.01T + 61T^{2} \) |
| 67 | \( 1 + 2.12T + 67T^{2} \) |
| 71 | \( 1 + 12.3T + 71T^{2} \) |
| 73 | \( 1 + 3.58T + 73T^{2} \) |
| 79 | \( 1 + 1.35T + 79T^{2} \) |
| 83 | \( 1 + 0.363T + 83T^{2} \) |
| 89 | \( 1 + 6.59T + 89T^{2} \) |
| 97 | \( 1 - 4.64T + 97T^{2} \) |
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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
−8.279653302558507365798040357782, −7.37757762871592390682407999598, −7.05523461570363272216859535678, −6.19479645548904560688124899797, −4.88141515718446060866109391485, −3.92282722400180165314815774589, −3.16816884638198704978965352467, −2.58017237512646593301748413052, −1.59532420836183380004308882251, 0,
1.59532420836183380004308882251, 2.58017237512646593301748413052, 3.16816884638198704978965352467, 3.92282722400180165314815774589, 4.88141515718446060866109391485, 6.19479645548904560688124899797, 7.05523461570363272216859535678, 7.37757762871592390682407999598, 8.279653302558507365798040357782
