| L(s) = 1 | + 2-s + 1.13·3-s + 4-s − 3.67·5-s + 1.13·6-s − 0.943·7-s + 8-s − 1.71·9-s − 3.67·10-s − 2.42·11-s + 1.13·12-s + 5.67·13-s − 0.943·14-s − 4.15·15-s + 16-s + 3.06·17-s − 1.71·18-s + 0.847·19-s − 3.67·20-s − 1.06·21-s − 2.42·22-s − 23-s + 1.13·24-s + 8.48·25-s + 5.67·26-s − 5.34·27-s − 0.943·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.653·3-s + 0.5·4-s − 1.64·5-s + 0.462·6-s − 0.356·7-s + 0.353·8-s − 0.572·9-s − 1.16·10-s − 0.730·11-s + 0.326·12-s + 1.57·13-s − 0.252·14-s − 1.07·15-s + 0.250·16-s + 0.744·17-s − 0.405·18-s + 0.194·19-s − 0.821·20-s − 0.233·21-s − 0.516·22-s − 0.208·23-s + 0.231·24-s + 1.69·25-s + 1.11·26-s − 1.02·27-s − 0.178·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| 131 | \( 1 - T \) |
| good | 3 | \( 1 - 1.13T + 3T^{2} \) |
| 5 | \( 1 + 3.67T + 5T^{2} \) |
| 7 | \( 1 + 0.943T + 7T^{2} \) |
| 11 | \( 1 + 2.42T + 11T^{2} \) |
| 13 | \( 1 - 5.67T + 13T^{2} \) |
| 17 | \( 1 - 3.06T + 17T^{2} \) |
| 19 | \( 1 - 0.847T + 19T^{2} \) |
| 29 | \( 1 - 7.36T + 29T^{2} \) |
| 31 | \( 1 - 2.78T + 31T^{2} \) |
| 37 | \( 1 + 7.52T + 37T^{2} \) |
| 41 | \( 1 - 2.56T + 41T^{2} \) |
| 43 | \( 1 + 8.93T + 43T^{2} \) |
| 47 | \( 1 + 2.98T + 47T^{2} \) |
| 53 | \( 1 + 1.01T + 53T^{2} \) |
| 59 | \( 1 + 9.86T + 59T^{2} \) |
| 61 | \( 1 + 5.95T + 61T^{2} \) |
| 67 | \( 1 + 3.67T + 67T^{2} \) |
| 71 | \( 1 - 8.79T + 71T^{2} \) |
| 73 | \( 1 + 9.83T + 73T^{2} \) |
| 79 | \( 1 + 13.1T + 79T^{2} \) |
| 83 | \( 1 - 7.24T + 83T^{2} \) |
| 89 | \( 1 + 13.6T + 89T^{2} \) |
| 97 | \( 1 + 0.0496T + 97T^{2} \) |
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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
−7.974730769905626460767894232714, −7.03796274321465815428712803404, −6.31852891549953938062150310503, −5.50935827203517347066871766655, −4.64586161307803128000418669514, −3.87213216761721836037547426023, −3.18031941388072063347770053831, −2.95587230738346624469120153491, −1.40255547290433352859370117012, 0,
1.40255547290433352859370117012, 2.95587230738346624469120153491, 3.18031941388072063347770053831, 3.87213216761721836037547426023, 4.64586161307803128000418669514, 5.50935827203517347066871766655, 6.31852891549953938062150310503, 7.03796274321465815428712803404, 7.974730769905626460767894232714
