L(s) = 1 | − 2.14·2-s − 2.87·3-s + 2.59·4-s − 5-s + 6.16·6-s + 3.10·7-s − 1.26·8-s + 5.28·9-s + 2.14·10-s − 1.10·11-s − 7.45·12-s + 1.77·13-s − 6.65·14-s + 2.87·15-s − 2.47·16-s + 7.75·17-s − 11.3·18-s + 19-s − 2.59·20-s − 8.93·21-s + 2.36·22-s − 6.65·23-s + 3.63·24-s + 25-s − 3.79·26-s − 6.57·27-s + 8.04·28-s + ⋯ |
L(s) = 1 | − 1.51·2-s − 1.66·3-s + 1.29·4-s − 0.447·5-s + 2.51·6-s + 1.17·7-s − 0.446·8-s + 1.76·9-s + 0.677·10-s − 0.333·11-s − 2.15·12-s + 0.491·13-s − 1.77·14-s + 0.743·15-s − 0.617·16-s + 1.88·17-s − 2.66·18-s + 0.229·19-s − 0.579·20-s − 1.95·21-s + 0.504·22-s − 1.38·23-s + 0.742·24-s + 0.200·25-s − 0.745·26-s − 1.26·27-s + 1.51·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3187981041\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3187981041\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + T \) |
| 19 | \( 1 - T \) |
good | 2 | \( 1 + 2.14T + 2T^{2} \) |
| 3 | \( 1 + 2.87T + 3T^{2} \) |
| 7 | \( 1 - 3.10T + 7T^{2} \) |
| 11 | \( 1 + 1.10T + 11T^{2} \) |
| 13 | \( 1 - 1.77T + 13T^{2} \) |
| 17 | \( 1 - 7.75T + 17T^{2} \) |
| 23 | \( 1 + 6.65T + 23T^{2} \) |
| 29 | \( 1 - 7.75T + 29T^{2} \) |
| 31 | \( 1 - 6.57T + 31T^{2} \) |
| 37 | \( 1 + 1.40T + 37T^{2} \) |
| 41 | \( 1 - 2.81T + 41T^{2} \) |
| 43 | \( 1 - 3.10T + 43T^{2} \) |
| 47 | \( 1 - 1.46T + 47T^{2} \) |
| 53 | \( 1 + 2.59T + 53T^{2} \) |
| 59 | \( 1 + 5.38T + 59T^{2} \) |
| 61 | \( 1 - 4.07T + 61T^{2} \) |
| 67 | \( 1 + 15.8T + 67T^{2} \) |
| 71 | \( 1 - 7.14T + 71T^{2} \) |
| 73 | \( 1 - 0.243T + 73T^{2} \) |
| 79 | \( 1 + 9.38T + 79T^{2} \) |
| 83 | \( 1 - 8.86T + 83T^{2} \) |
| 89 | \( 1 - 0.813T + 89T^{2} \) |
| 97 | \( 1 - 3.19T + 97T^{2} \) |
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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
−14.08867392670944286415376980324, −12.17540404242979426924189436858, −11.63437178223277940049474798000, −10.65042837876770006756195843618, −10.01632690744810134929308459940, −8.271077743302847480931738654135, −7.53692344142704791923134165805, −6.05966991595158068268064220334, −4.71140324737967169213189313044, −1.10811879022495186699818444024,
1.10811879022495186699818444024, 4.71140324737967169213189313044, 6.05966991595158068268064220334, 7.53692344142704791923134165805, 8.271077743302847480931738654135, 10.01632690744810134929308459940, 10.65042837876770006756195843618, 11.63437178223277940049474798000, 12.17540404242979426924189436858, 14.08867392670944286415376980324