| L(s) = 1 | − 1.93·2-s − 0.393·3-s + 1.75·4-s + 0.762·6-s − 4.57·7-s + 0.481·8-s − 2.84·9-s − 2.65·11-s − 0.689·12-s + 6.10·13-s + 8.85·14-s − 4.43·16-s − 3.76·17-s + 5.51·18-s − 1.25·19-s + 1.79·21-s + 5.14·22-s + 8.79·23-s − 0.189·24-s − 11.8·26-s + 2.30·27-s − 8.00·28-s − 3.88·29-s − 4.36·31-s + 7.62·32-s + 1.04·33-s + 7.29·34-s + ⋯ |
| L(s) = 1 | − 1.36·2-s − 0.227·3-s + 0.875·4-s + 0.311·6-s − 1.72·7-s + 0.170·8-s − 0.948·9-s − 0.801·11-s − 0.198·12-s + 1.69·13-s + 2.36·14-s − 1.10·16-s − 0.913·17-s + 1.29·18-s − 0.288·19-s + 0.392·21-s + 1.09·22-s + 1.83·23-s − 0.0387·24-s − 2.31·26-s + 0.442·27-s − 1.51·28-s − 0.720·29-s − 0.783·31-s + 1.34·32-s + 0.182·33-s + 1.25·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 197 | \( 1 - T \) |
| good | 2 | \( 1 + 1.93T + 2T^{2} \) |
| 3 | \( 1 + 0.393T + 3T^{2} \) |
| 7 | \( 1 + 4.57T + 7T^{2} \) |
| 11 | \( 1 + 2.65T + 11T^{2} \) |
| 13 | \( 1 - 6.10T + 13T^{2} \) |
| 17 | \( 1 + 3.76T + 17T^{2} \) |
| 19 | \( 1 + 1.25T + 19T^{2} \) |
| 23 | \( 1 - 8.79T + 23T^{2} \) |
| 29 | \( 1 + 3.88T + 29T^{2} \) |
| 31 | \( 1 + 4.36T + 31T^{2} \) |
| 37 | \( 1 - 2.74T + 37T^{2} \) |
| 41 | \( 1 - 2.33T + 41T^{2} \) |
| 43 | \( 1 + 6.44T + 43T^{2} \) |
| 47 | \( 1 - 0.0382T + 47T^{2} \) |
| 53 | \( 1 - 5.36T + 53T^{2} \) |
| 59 | \( 1 + 4.00T + 59T^{2} \) |
| 61 | \( 1 - 14.5T + 61T^{2} \) |
| 67 | \( 1 - 13.5T + 67T^{2} \) |
| 71 | \( 1 + 0.772T + 71T^{2} \) |
| 73 | \( 1 - 11.9T + 73T^{2} \) |
| 79 | \( 1 + 14.2T + 79T^{2} \) |
| 83 | \( 1 + 10.2T + 83T^{2} \) |
| 89 | \( 1 - 5.99T + 89T^{2} \) |
| 97 | \( 1 + 12.8T + 97T^{2} \) |
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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
−8.226941250497167976600033414437, −7.11678377184883781104172684956, −6.68586192240100697051777816328, −5.96217595274541023465402628428, −5.19543602572875450510220208611, −3.91408077849314882809727692623, −3.12786162217130342047456352035, −2.28699540722660831141221637238, −0.878802522521460609329094736442, 0,
0.878802522521460609329094736442, 2.28699540722660831141221637238, 3.12786162217130342047456352035, 3.91408077849314882809727692623, 5.19543602572875450510220208611, 5.96217595274541023465402628428, 6.68586192240100697051777816328, 7.11678377184883781104172684956, 8.226941250497167976600033414437
