| L(s) = 1 | + 0.0849·2-s + 1.30·3-s − 1.99·4-s + 0.110·6-s − 0.925·7-s − 0.339·8-s − 1.30·9-s + 3.08·11-s − 2.59·12-s − 0.928·13-s − 0.0786·14-s + 3.95·16-s − 0.233·17-s − 0.110·18-s + 4.41·19-s − 1.20·21-s + 0.262·22-s + 0.167·23-s − 0.442·24-s − 0.0788·26-s − 5.60·27-s + 1.84·28-s − 7.54·29-s − 8.34·31-s + 1.01·32-s + 4.01·33-s − 0.0198·34-s + ⋯ |
| L(s) = 1 | + 0.0600·2-s + 0.752·3-s − 0.996·4-s + 0.0452·6-s − 0.349·7-s − 0.119·8-s − 0.434·9-s + 0.929·11-s − 0.749·12-s − 0.257·13-s − 0.0210·14-s + 0.989·16-s − 0.0567·17-s − 0.0260·18-s + 1.01·19-s − 0.263·21-s + 0.0558·22-s + 0.0350·23-s − 0.0902·24-s − 0.0154·26-s − 1.07·27-s + 0.348·28-s − 1.40·29-s − 1.49·31-s + 0.179·32-s + 0.699·33-s − 0.00340·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 241 | \( 1 + T \) |
| good | 2 | \( 1 - 0.0849T + 2T^{2} \) |
| 3 | \( 1 - 1.30T + 3T^{2} \) |
| 7 | \( 1 + 0.925T + 7T^{2} \) |
| 11 | \( 1 - 3.08T + 11T^{2} \) |
| 13 | \( 1 + 0.928T + 13T^{2} \) |
| 17 | \( 1 + 0.233T + 17T^{2} \) |
| 19 | \( 1 - 4.41T + 19T^{2} \) |
| 23 | \( 1 - 0.167T + 23T^{2} \) |
| 29 | \( 1 + 7.54T + 29T^{2} \) |
| 31 | \( 1 + 8.34T + 31T^{2} \) |
| 37 | \( 1 - 9.22T + 37T^{2} \) |
| 41 | \( 1 - 2.33T + 41T^{2} \) |
| 43 | \( 1 - 6.63T + 43T^{2} \) |
| 47 | \( 1 + 10.0T + 47T^{2} \) |
| 53 | \( 1 - 4.19T + 53T^{2} \) |
| 59 | \( 1 + 2.28T + 59T^{2} \) |
| 61 | \( 1 - 11.2T + 61T^{2} \) |
| 67 | \( 1 + 6.13T + 67T^{2} \) |
| 71 | \( 1 + 8.46T + 71T^{2} \) |
| 73 | \( 1 - 5.93T + 73T^{2} \) |
| 79 | \( 1 + 5.88T + 79T^{2} \) |
| 83 | \( 1 - 3.10T + 83T^{2} \) |
| 89 | \( 1 + 10.3T + 89T^{2} \) |
| 97 | \( 1 - 8.84T + 97T^{2} \) |
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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
−7.74381414089783361154082811827, −7.28966051316279539695918889887, −6.13409886872717177858447564053, −5.59679962759631403059636426858, −4.75383920232185450422553955856, −3.79155956715781320940741413048, −3.46615077446636326146733001568, −2.46996077545237710982589656618, −1.30135245085062972735155306577, 0,
1.30135245085062972735155306577, 2.46996077545237710982589656618, 3.46615077446636326146733001568, 3.79155956715781320940741413048, 4.75383920232185450422553955856, 5.59679962759631403059636426858, 6.13409886872717177858447564053, 7.28966051316279539695918889887, 7.74381414089783361154082811827
