| L(s) = 1 | − 0.183·2-s − 1.96·4-s + 3.26·5-s − 0.215·7-s + 0.726·8-s − 0.597·10-s − 2.92·11-s − 3.58·13-s + 0.0394·14-s + 3.79·16-s + 7.11·17-s + 1.38·19-s − 6.41·20-s + 0.535·22-s + 6.71·23-s + 5.63·25-s + 0.656·26-s + 0.423·28-s + 6.41·31-s − 2.14·32-s − 1.30·34-s − 0.702·35-s − 6.11·37-s − 0.253·38-s + 2.36·40-s + 6.50·41-s − 6.24·43-s + ⋯ |
| L(s) = 1 | − 0.129·2-s − 0.983·4-s + 1.45·5-s − 0.0813·7-s + 0.256·8-s − 0.188·10-s − 0.881·11-s − 0.994·13-s + 0.0105·14-s + 0.949·16-s + 1.72·17-s + 0.317·19-s − 1.43·20-s + 0.114·22-s + 1.40·23-s + 1.12·25-s + 0.128·26-s + 0.0800·28-s + 1.15·31-s − 0.379·32-s − 0.223·34-s − 0.118·35-s − 1.00·37-s − 0.0410·38-s + 0.374·40-s + 1.01·41-s − 0.952·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7569 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7569 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.917379213\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.917379213\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 29 | \( 1 \) |
| good | 2 | \( 1 + 0.183T + 2T^{2} \) |
| 5 | \( 1 - 3.26T + 5T^{2} \) |
| 7 | \( 1 + 0.215T + 7T^{2} \) |
| 11 | \( 1 + 2.92T + 11T^{2} \) |
| 13 | \( 1 + 3.58T + 13T^{2} \) |
| 17 | \( 1 - 7.11T + 17T^{2} \) |
| 19 | \( 1 - 1.38T + 19T^{2} \) |
| 23 | \( 1 - 6.71T + 23T^{2} \) |
| 31 | \( 1 - 6.41T + 31T^{2} \) |
| 37 | \( 1 + 6.11T + 37T^{2} \) |
| 41 | \( 1 - 6.50T + 41T^{2} \) |
| 43 | \( 1 + 6.24T + 43T^{2} \) |
| 47 | \( 1 - 9.04T + 47T^{2} \) |
| 53 | \( 1 + 4.53T + 53T^{2} \) |
| 59 | \( 1 + 12.0T + 59T^{2} \) |
| 61 | \( 1 - 1.58T + 61T^{2} \) |
| 67 | \( 1 + 7.18T + 67T^{2} \) |
| 71 | \( 1 - 13.4T + 71T^{2} \) |
| 73 | \( 1 + 13.8T + 73T^{2} \) |
| 79 | \( 1 + 3.35T + 79T^{2} \) |
| 83 | \( 1 - 5.23T + 83T^{2} \) |
| 89 | \( 1 - 12.9T + 89T^{2} \) |
| 97 | \( 1 - 3.92T + 97T^{2} \) |
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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
−7.80834662326833449050351636504, −7.39498508216891641195528104096, −6.32815192246337735143905201295, −5.62016124008237595646927483670, −5.10015249548206518087687174302, −4.67541631078150860543762943081, −3.30658448986381622381857510957, −2.78854821328600252291656389107, −1.67368061418384315615505288318, −0.74042989821156927320148775086,
0.74042989821156927320148775086, 1.67368061418384315615505288318, 2.78854821328600252291656389107, 3.30658448986381622381857510957, 4.67541631078150860543762943081, 5.10015249548206518087687174302, 5.62016124008237595646927483670, 6.32815192246337735143905201295, 7.39498508216891641195528104096, 7.80834662326833449050351636504
