| L(s) = 1 | + 2-s − 2.65·3-s + 4-s + 0.347·5-s − 2.65·6-s − 2.28·7-s + 8-s + 4.03·9-s + 0.347·10-s + 0.505·11-s − 2.65·12-s − 4.28·13-s − 2.28·14-s − 0.920·15-s + 16-s + 3.45·17-s + 4.03·18-s + 0.543·19-s + 0.347·20-s + 6.06·21-s + 0.505·22-s + 23-s − 2.65·24-s − 4.87·25-s − 4.28·26-s − 2.73·27-s − 2.28·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.53·3-s + 0.5·4-s + 0.155·5-s − 1.08·6-s − 0.864·7-s + 0.353·8-s + 1.34·9-s + 0.109·10-s + 0.152·11-s − 0.765·12-s − 1.18·13-s − 0.611·14-s − 0.237·15-s + 0.250·16-s + 0.837·17-s + 0.950·18-s + 0.124·19-s + 0.0775·20-s + 1.32·21-s + 0.107·22-s + 0.208·23-s − 0.541·24-s − 0.975·25-s − 0.839·26-s − 0.526·27-s − 0.432·28-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 | \( 1 + 2.65T + 3T^{2} \) |
| 5 | \( 1 - 0.347T + 5T^{2} \) |
| 7 | \( 1 + 2.28T + 7T^{2} \) |
| 11 | \( 1 - 0.505T + 11T^{2} \) |
| 13 | \( 1 + 4.28T + 13T^{2} \) |
| 17 | \( 1 - 3.45T + 17T^{2} \) |
| 19 | \( 1 - 0.543T + 19T^{2} \) |
| 29 | \( 1 - 1.89T + 29T^{2} \) |
| 31 | \( 1 - 7.05T + 31T^{2} \) |
| 37 | \( 1 + 0.755T + 37T^{2} \) |
| 41 | \( 1 + 3.09T + 41T^{2} \) |
| 43 | \( 1 - 4.47T + 43T^{2} \) |
| 47 | \( 1 - 0.558T + 47T^{2} \) |
| 53 | \( 1 + 1.57T + 53T^{2} \) |
| 59 | \( 1 - 7.64T + 59T^{2} \) |
| 61 | \( 1 + 4.52T + 61T^{2} \) |
| 67 | \( 1 - 3.89T + 67T^{2} \) |
| 71 | \( 1 + 9.51T + 71T^{2} \) |
| 73 | \( 1 - 12.4T + 73T^{2} \) |
| 79 | \( 1 + 17.0T + 79T^{2} \) |
| 83 | \( 1 + 3.06T + 83T^{2} \) |
| 89 | \( 1 - 0.159T + 89T^{2} \) |
| 97 | \( 1 - 7.33T + 97T^{2} \) |
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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
−7.33180548548914028331212358970, −6.79095076395573453868287221975, −6.13123309738771536833688739432, −5.62674181422312473304206935612, −4.95554181723613642299269659692, −4.30861977595753655429333019625, −3.32659597157574367987151825049, −2.45221599161474512394726548960, −1.15583835859046667761759441228, 0,
1.15583835859046667761759441228, 2.45221599161474512394726548960, 3.32659597157574367987151825049, 4.30861977595753655429333019625, 4.95554181723613642299269659692, 5.62674181422312473304206935612, 6.13123309738771536833688739432, 6.79095076395573453868287221975, 7.33180548548914028331212358970
