| L(s) = 1 | + 1.22·2-s − 3-s − 0.504·4-s − 1.22·6-s + 4.18·7-s − 3.06·8-s + 9-s + 1.89·11-s + 0.504·12-s − 13-s + 5.11·14-s − 2.73·16-s − 1.73·17-s + 1.22·18-s − 1.11·19-s − 4.18·21-s + 2.32·22-s + 9.30·23-s + 3.06·24-s − 1.22·26-s − 27-s − 2.10·28-s + 5.00·29-s + 10.0·31-s + 2.77·32-s − 1.89·33-s − 2.12·34-s + ⋯ |
| L(s) = 1 | + 0.864·2-s − 0.577·3-s − 0.252·4-s − 0.499·6-s + 1.58·7-s − 1.08·8-s + 0.333·9-s + 0.572·11-s + 0.145·12-s − 0.277·13-s + 1.36·14-s − 0.684·16-s − 0.421·17-s + 0.288·18-s − 0.256·19-s − 0.912·21-s + 0.494·22-s + 1.93·23-s + 0.625·24-s − 0.239·26-s − 0.192·27-s − 0.398·28-s + 0.929·29-s + 1.79·31-s + 0.490·32-s − 0.330·33-s − 0.364·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.095392346\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.095392346\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 - 1.22T + 2T^{2} \) |
| 7 | \( 1 - 4.18T + 7T^{2} \) |
| 11 | \( 1 - 1.89T + 11T^{2} \) |
| 17 | \( 1 + 1.73T + 17T^{2} \) |
| 19 | \( 1 + 1.11T + 19T^{2} \) |
| 23 | \( 1 - 9.30T + 23T^{2} \) |
| 29 | \( 1 - 5.00T + 29T^{2} \) |
| 31 | \( 1 - 10.0T + 31T^{2} \) |
| 37 | \( 1 + 11.3T + 37T^{2} \) |
| 41 | \( 1 - 8.02T + 41T^{2} \) |
| 43 | \( 1 - 1.00T + 43T^{2} \) |
| 47 | \( 1 - 5.63T + 47T^{2} \) |
| 53 | \( 1 - 0.174T + 53T^{2} \) |
| 59 | \( 1 - 8.64T + 59T^{2} \) |
| 61 | \( 1 - 3.27T + 61T^{2} \) |
| 67 | \( 1 + 11.4T + 67T^{2} \) |
| 71 | \( 1 + 5.37T + 71T^{2} \) |
| 73 | \( 1 + 4.01T + 73T^{2} \) |
| 79 | \( 1 + 0.613T + 79T^{2} \) |
| 83 | \( 1 + 9.08T + 83T^{2} \) |
| 89 | \( 1 + 1.46T + 89T^{2} \) |
| 97 | \( 1 - 3.85T + 97T^{2} \) |
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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
−10.21396475853284592721227968332, −8.970814350442073243322314568455, −8.503050396858661648606736607530, −7.29159895497517644304888727188, −6.41386530714260979309743240774, −5.35507587830955301025933616676, −4.76156920693138462606872133359, −4.15361300008600495772518526955, −2.70845096134043450493200298941, −1.12566635681067910688242264285,
1.12566635681067910688242264285, 2.70845096134043450493200298941, 4.15361300008600495772518526955, 4.76156920693138462606872133359, 5.35507587830955301025933616676, 6.41386530714260979309743240774, 7.29159895497517644304888727188, 8.503050396858661648606736607530, 8.970814350442073243322314568455, 10.21396475853284592721227968332
