| L(s) = 1 | + 2.49·2-s + 4.23·4-s + 3.08·7-s + 5.58·8-s − 3·9-s + 1.90·13-s + 7.70·14-s + 5.47·16-s + 8.08·17-s − 7.49·18-s + 4.76·26-s + 13.0·28-s − 8.94·31-s + 2.49·32-s + 20.1·34-s − 12.7·36-s + 13.0·43-s + 2.52·49-s + 8.08·52-s + 17.2·56-s + 4·59-s − 22.3·62-s − 9.26·63-s − 4.70·64-s + 34.2·68-s + 8·71-s − 16.7·72-s + ⋯ |
| L(s) = 1 | + 1.76·2-s + 2.11·4-s + 1.16·7-s + 1.97·8-s − 9-s + 0.529·13-s + 2.06·14-s + 1.36·16-s + 1.95·17-s − 1.76·18-s + 0.934·26-s + 2.47·28-s − 1.60·31-s + 0.441·32-s + 3.46·34-s − 2.11·36-s + 1.99·43-s + 0.361·49-s + 1.12·52-s + 2.30·56-s + 0.520·59-s − 2.83·62-s − 1.16·63-s − 0.588·64-s + 4.15·68-s + 0.949·71-s − 1.97·72-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.322184180\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.322184180\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 2.49T + 2T^{2} \) |
| 3 | \( 1 + 3T^{2} \) |
| 7 | \( 1 - 3.08T + 7T^{2} \) |
| 13 | \( 1 - 1.90T + 13T^{2} \) |
| 17 | \( 1 - 8.08T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 8.94T + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 13.0T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 + 11.8T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 0.728T + 83T^{2} \) |
| 89 | \( 1 + 13.4T + 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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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
−8.473291660016968492768407674674, −7.76340479519632558353133458701, −7.11432245015152368475284280057, −5.88649951389735887708368698058, −5.64219923129182479281696422672, −4.96033068618829110738026902472, −3.99854053401464183553525482642, −3.33080644616776745638876018303, −2.43170646565422917163681922543, −1.35017683973243630458672413125,
1.35017683973243630458672413125, 2.43170646565422917163681922543, 3.33080644616776745638876018303, 3.99854053401464183553525482642, 4.96033068618829110738026902472, 5.64219923129182479281696422672, 5.88649951389735887708368698058, 7.11432245015152368475284280057, 7.76340479519632558353133458701, 8.473291660016968492768407674674
