| L(s) = 1 | − 2-s + 4-s + 3.87·5-s − 4.40·7-s − 8-s − 3.87·10-s − 1.27·11-s + 2.79·13-s + 4.40·14-s + 16-s + 0.353·17-s + 4.03·19-s + 3.87·20-s + 1.27·22-s − 3.58·23-s + 10.0·25-s − 2.79·26-s − 4.40·28-s − 0.972·29-s − 9.96·31-s − 32-s − 0.353·34-s − 17.1·35-s − 6.22·37-s − 4.03·38-s − 3.87·40-s − 7.05·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 1.73·5-s − 1.66·7-s − 0.353·8-s − 1.22·10-s − 0.383·11-s + 0.775·13-s + 1.17·14-s + 0.250·16-s + 0.0856·17-s + 0.924·19-s + 0.867·20-s + 0.270·22-s − 0.747·23-s + 2.00·25-s − 0.548·26-s − 0.833·28-s − 0.180·29-s − 1.78·31-s − 0.176·32-s − 0.0605·34-s − 2.89·35-s − 1.02·37-s − 0.653·38-s − 0.613·40-s − 1.10·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6354 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 353 | \( 1 - T \) |
| good | 5 | \( 1 - 3.87T + 5T^{2} \) |
| 7 | \( 1 + 4.40T + 7T^{2} \) |
| 11 | \( 1 + 1.27T + 11T^{2} \) |
| 13 | \( 1 - 2.79T + 13T^{2} \) |
| 17 | \( 1 - 0.353T + 17T^{2} \) |
| 19 | \( 1 - 4.03T + 19T^{2} \) |
| 23 | \( 1 + 3.58T + 23T^{2} \) |
| 29 | \( 1 + 0.972T + 29T^{2} \) |
| 31 | \( 1 + 9.96T + 31T^{2} \) |
| 37 | \( 1 + 6.22T + 37T^{2} \) |
| 41 | \( 1 + 7.05T + 41T^{2} \) |
| 43 | \( 1 + 3.73T + 43T^{2} \) |
| 47 | \( 1 - 9.84T + 47T^{2} \) |
| 53 | \( 1 + 13.6T + 53T^{2} \) |
| 59 | \( 1 + 3.11T + 59T^{2} \) |
| 61 | \( 1 - 7.07T + 61T^{2} \) |
| 67 | \( 1 - 11.6T + 67T^{2} \) |
| 71 | \( 1 + 14.1T + 71T^{2} \) |
| 73 | \( 1 + 7.95T + 73T^{2} \) |
| 79 | \( 1 + 8.78T + 79T^{2} \) |
| 83 | \( 1 - 5.07T + 83T^{2} \) |
| 89 | \( 1 + 2.90T + 89T^{2} \) |
| 97 | \( 1 - 17.8T + 97T^{2} \) |
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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
−7.57424297295489668948713015685, −6.89773937109943430827097668186, −6.25577145773312295370918823596, −5.78246206578440514172920668197, −5.19197079023740090599252248872, −3.64692062713635145522542330441, −3.08109513356421347461881959861, −2.15995354196765430405886592652, −1.37021429100059542442743736121, 0,
1.37021429100059542442743736121, 2.15995354196765430405886592652, 3.08109513356421347461881959861, 3.64692062713635145522542330441, 5.19197079023740090599252248872, 5.78246206578440514172920668197, 6.25577145773312295370918823596, 6.89773937109943430827097668186, 7.57424297295489668948713015685
