| L(s) = 1 | − 2.10·2-s + 1.86·3-s + 2.42·4-s − 5-s − 3.92·6-s − 0.899·8-s + 0.483·9-s + 2.10·10-s + 11-s + 4.53·12-s − 4.19·13-s − 1.86·15-s − 2.96·16-s − 6.50·17-s − 1.01·18-s − 3.44·19-s − 2.42·20-s − 2.10·22-s − 3.61·23-s − 1.67·24-s + 25-s + 8.83·26-s − 4.69·27-s + 6.70·29-s + 3.92·30-s + 4.42·31-s + 8.03·32-s + ⋯ |
| L(s) = 1 | − 1.48·2-s + 1.07·3-s + 1.21·4-s − 0.447·5-s − 1.60·6-s − 0.317·8-s + 0.161·9-s + 0.665·10-s + 0.301·11-s + 1.30·12-s − 1.16·13-s − 0.481·15-s − 0.740·16-s − 1.57·17-s − 0.239·18-s − 0.790·19-s − 0.542·20-s − 0.448·22-s − 0.753·23-s − 0.342·24-s + 0.200·25-s + 1.73·26-s − 0.903·27-s + 1.24·29-s + 0.717·30-s + 0.795·31-s + 1.41·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2695 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2695 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8284199571\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8284199571\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 + 2.10T + 2T^{2} \) |
| 3 | \( 1 - 1.86T + 3T^{2} \) |
| 13 | \( 1 + 4.19T + 13T^{2} \) |
| 17 | \( 1 + 6.50T + 17T^{2} \) |
| 19 | \( 1 + 3.44T + 19T^{2} \) |
| 23 | \( 1 + 3.61T + 23T^{2} \) |
| 29 | \( 1 - 6.70T + 29T^{2} \) |
| 31 | \( 1 - 4.42T + 31T^{2} \) |
| 37 | \( 1 - 6.51T + 37T^{2} \) |
| 41 | \( 1 - 4.37T + 41T^{2} \) |
| 43 | \( 1 - 9.77T + 43T^{2} \) |
| 47 | \( 1 - 3.08T + 47T^{2} \) |
| 53 | \( 1 - 13.3T + 53T^{2} \) |
| 59 | \( 1 - 8.72T + 59T^{2} \) |
| 61 | \( 1 + 3.73T + 61T^{2} \) |
| 67 | \( 1 + 4.60T + 67T^{2} \) |
| 71 | \( 1 - 10.4T + 71T^{2} \) |
| 73 | \( 1 - 6.66T + 73T^{2} \) |
| 79 | \( 1 + 3.30T + 79T^{2} \) |
| 83 | \( 1 + 3.04T + 83T^{2} \) |
| 89 | \( 1 - 17.4T + 89T^{2} \) |
| 97 | \( 1 - 10.5T + 97T^{2} \) |
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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
−8.818148497617759857965146335964, −8.266138013418836502588349727100, −7.66101997680202825537624296235, −6.97856928186150030067797162741, −6.17154706108378516315746171052, −4.62584903191689243288913844939, −4.02773529992054542587568839261, −2.48407324838841402262585263612, −2.30525023663701723669125918239, −0.64544575002769739120705917681,
0.64544575002769739120705917681, 2.30525023663701723669125918239, 2.48407324838841402262585263612, 4.02773529992054542587568839261, 4.62584903191689243288913844939, 6.17154706108378516315746171052, 6.97856928186150030067797162741, 7.66101997680202825537624296235, 8.266138013418836502588349727100, 8.818148497617759857965146335964
