| L(s) = 1 | + 2.39·2-s + 3.73·4-s + 0.732·7-s + 4.14·8-s + 6.54·11-s − 3.46·13-s + 1.75·14-s + 2.46·16-s + 0.641·17-s − 6.73·19-s + 15.6·22-s + 3.50·23-s − 8.29·26-s + 2.73·28-s + 7.18·29-s + 31-s − 2.39·32-s + 1.53·34-s + 2.73·37-s − 16.1·38-s + 11.3·41-s + 6.46·43-s + 24.4·44-s + 8.39·46-s + 10.6·47-s − 6.46·49-s − 12.9·52-s + ⋯ |
| L(s) = 1 | + 1.69·2-s + 1.86·4-s + 0.276·7-s + 1.46·8-s + 1.97·11-s − 0.960·13-s + 0.468·14-s + 0.616·16-s + 0.155·17-s − 1.54·19-s + 3.33·22-s + 0.730·23-s − 1.62·26-s + 0.516·28-s + 1.33·29-s + 0.179·31-s − 0.423·32-s + 0.263·34-s + 0.449·37-s − 2.61·38-s + 1.76·41-s + 0.985·43-s + 3.68·44-s + 1.23·46-s + 1.55·47-s − 0.923·49-s − 1.79·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.712397400\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.712397400\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 - 2.39T + 2T^{2} \) |
| 7 | \( 1 - 0.732T + 7T^{2} \) |
| 11 | \( 1 - 6.54T + 11T^{2} \) |
| 13 | \( 1 + 3.46T + 13T^{2} \) |
| 17 | \( 1 - 0.641T + 17T^{2} \) |
| 19 | \( 1 + 6.73T + 19T^{2} \) |
| 23 | \( 1 - 3.50T + 23T^{2} \) |
| 29 | \( 1 - 7.18T + 29T^{2} \) |
| 37 | \( 1 - 2.73T + 37T^{2} \) |
| 41 | \( 1 - 11.3T + 41T^{2} \) |
| 43 | \( 1 - 6.46T + 43T^{2} \) |
| 47 | \( 1 - 10.6T + 47T^{2} \) |
| 53 | \( 1 - 11.3T + 53T^{2} \) |
| 59 | \( 1 + 6.54T + 59T^{2} \) |
| 61 | \( 1 + 4.19T + 61T^{2} \) |
| 67 | \( 1 + 7.66T + 67T^{2} \) |
| 71 | \( 1 - 4.14T + 71T^{2} \) |
| 73 | \( 1 - 10T + 73T^{2} \) |
| 79 | \( 1 + 5.53T + 79T^{2} \) |
| 83 | \( 1 + 13.0T + 83T^{2} \) |
| 89 | \( 1 + 7.65T + 89T^{2} \) |
| 97 | \( 1 - 13.7T + 97T^{2} \) |
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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
−7.57031480001624292872764714290, −6.97958024346710919388669674826, −6.29969582076965131037621524225, −5.91119133183234036856461805678, −4.85816856077895845725644042194, −4.32480836100496370137536856720, −3.92893547935690755662160219506, −2.84762867819482174571732581900, −2.21557361908663187561434648072, −1.07992475728170579034973280537,
1.07992475728170579034973280537, 2.21557361908663187561434648072, 2.84762867819482174571732581900, 3.92893547935690755662160219506, 4.32480836100496370137536856720, 4.85816856077895845725644042194, 5.91119133183234036856461805678, 6.29969582076965131037621524225, 6.97958024346710919388669674826, 7.57031480001624292872764714290
