| L(s) = 1 | − 1.46·2-s + 3-s + 0.139·4-s − 1.46·6-s + 1.32·7-s + 2.72·8-s + 9-s − 5.32·11-s + 0.139·12-s − 2.39·13-s − 1.93·14-s − 4.25·16-s + 5.04·17-s − 1.46·18-s + 0.925·19-s + 1.32·21-s + 7.78·22-s − 5.72·23-s + 2.72·24-s + 3.50·26-s + 27-s + 0.184·28-s + 29-s + 5.72·31-s + 0.786·32-s − 5.32·33-s − 7.37·34-s + ⋯ |
| L(s) = 1 | − 1.03·2-s + 0.577·3-s + 0.0695·4-s − 0.597·6-s + 0.500·7-s + 0.962·8-s + 0.333·9-s − 1.60·11-s + 0.0401·12-s − 0.665·13-s − 0.517·14-s − 1.06·16-s + 1.22·17-s − 0.344·18-s + 0.212·19-s + 0.288·21-s + 1.65·22-s − 1.19·23-s + 0.555·24-s + 0.687·26-s + 0.192·27-s + 0.0348·28-s + 0.185·29-s + 1.02·31-s + 0.138·32-s − 0.926·33-s − 1.26·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 + 1.46T + 2T^{2} \) |
| 7 | \( 1 - 1.32T + 7T^{2} \) |
| 11 | \( 1 + 5.32T + 11T^{2} \) |
| 13 | \( 1 + 2.39T + 13T^{2} \) |
| 17 | \( 1 - 5.04T + 17T^{2} \) |
| 19 | \( 1 - 0.925T + 19T^{2} \) |
| 23 | \( 1 + 5.72T + 23T^{2} \) |
| 31 | \( 1 - 5.72T + 31T^{2} \) |
| 37 | \( 1 + 1.07T + 37T^{2} \) |
| 41 | \( 1 + 11.2T + 41T^{2} \) |
| 43 | \( 1 + 9.29T + 43T^{2} \) |
| 47 | \( 1 + 3.97T + 47T^{2} \) |
| 53 | \( 1 + 2.12T + 53T^{2} \) |
| 59 | \( 1 + 1.35T + 59T^{2} \) |
| 61 | \( 1 - 2.92T + 61T^{2} \) |
| 67 | \( 1 - 7.97T + 67T^{2} \) |
| 71 | \( 1 + 4.12T + 71T^{2} \) |
| 73 | \( 1 - 1.07T + 73T^{2} \) |
| 79 | \( 1 - 8.36T + 79T^{2} \) |
| 83 | \( 1 - 4.79T + 83T^{2} \) |
| 89 | \( 1 + 0.547T + 89T^{2} \) |
| 97 | \( 1 - 5.57T + 97T^{2} \) |
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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
−8.431305038387335236144784036919, −8.020941212731803498981576918989, −7.67488377894685106830099142240, −6.64539560275474378142995615595, −5.23722153596370632844623904658, −4.84166293738482365120223238417, −3.55969777565999339269089988445, −2.49006891216790608009569272847, −1.49509356174790438486979206418, 0,
1.49509356174790438486979206418, 2.49006891216790608009569272847, 3.55969777565999339269089988445, 4.84166293738482365120223238417, 5.23722153596370632844623904658, 6.64539560275474378142995615595, 7.67488377894685106830099142240, 8.020941212731803498981576918989, 8.431305038387335236144784036919
