| L(s) = 1 | − 1.69·2-s + 1.96·3-s + 0.887·4-s − 5-s − 3.33·6-s + 1.89·8-s + 0.848·9-s + 1.69·10-s − 0.494·11-s + 1.74·12-s − 1.20·13-s − 1.96·15-s − 4.98·16-s + 3.71·17-s − 1.44·18-s + 2.28·19-s − 0.887·20-s + 0.840·22-s − 2.66·23-s + 3.70·24-s + 25-s + 2.04·26-s − 4.22·27-s + 1.12·29-s + 3.33·30-s − 0.433·31-s + 4.69·32-s + ⋯ |
| L(s) = 1 | − 1.20·2-s + 1.13·3-s + 0.443·4-s − 0.447·5-s − 1.36·6-s + 0.668·8-s + 0.282·9-s + 0.537·10-s − 0.149·11-s + 0.502·12-s − 0.333·13-s − 0.506·15-s − 1.24·16-s + 0.900·17-s − 0.339·18-s + 0.523·19-s − 0.198·20-s + 0.179·22-s − 0.554·23-s + 0.757·24-s + 0.200·25-s + 0.400·26-s − 0.812·27-s + 0.209·29-s + 0.608·30-s − 0.0779·31-s + 0.829·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9065 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9065 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 2 | \( 1 + 1.69T + 2T^{2} \) |
| 3 | \( 1 - 1.96T + 3T^{2} \) |
| 11 | \( 1 + 0.494T + 11T^{2} \) |
| 13 | \( 1 + 1.20T + 13T^{2} \) |
| 17 | \( 1 - 3.71T + 17T^{2} \) |
| 19 | \( 1 - 2.28T + 19T^{2} \) |
| 23 | \( 1 + 2.66T + 23T^{2} \) |
| 29 | \( 1 - 1.12T + 29T^{2} \) |
| 31 | \( 1 + 0.433T + 31T^{2} \) |
| 41 | \( 1 + 3.97T + 41T^{2} \) |
| 43 | \( 1 - 5.04T + 43T^{2} \) |
| 47 | \( 1 + 0.141T + 47T^{2} \) |
| 53 | \( 1 - 5.17T + 53T^{2} \) |
| 59 | \( 1 + 10.3T + 59T^{2} \) |
| 61 | \( 1 + 0.146T + 61T^{2} \) |
| 67 | \( 1 + 11.8T + 67T^{2} \) |
| 71 | \( 1 + 6.46T + 71T^{2} \) |
| 73 | \( 1 - 2.16T + 73T^{2} \) |
| 79 | \( 1 - 2.22T + 79T^{2} \) |
| 83 | \( 1 - 6.75T + 83T^{2} \) |
| 89 | \( 1 - 10.5T + 89T^{2} \) |
| 97 | \( 1 + 3.48T + 97T^{2} \) |
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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
−7.54110891533112260078891683803, −7.33900083977661000477774637901, −6.21725510968246997867081276808, −5.28780479904575790627167550709, −4.44964858989860269844779366803, −3.63987699321442772429816885363, −2.92987593082017553643175462322, −2.08292919411977492778004019886, −1.17036317280734157944903683203, 0,
1.17036317280734157944903683203, 2.08292919411977492778004019886, 2.92987593082017553643175462322, 3.63987699321442772429816885363, 4.44964858989860269844779366803, 5.28780479904575790627167550709, 6.21725510968246997867081276808, 7.33900083977661000477774637901, 7.54110891533112260078891683803
