| L(s) = 1 | + 1.88·5-s + 7·7-s + 52.4·11-s + 4.64·13-s − 98.1·17-s + 110.·19-s + 84.2·23-s − 121.·25-s + 112.·29-s + 42.0·31-s + 13.1·35-s − 392.·37-s + 235.·41-s + 84.4·43-s − 39.8·47-s + 49·49-s + 178.·53-s + 98.8·55-s − 2.29·59-s + 768.·61-s + 8.75·65-s + 224.·67-s + 780.·71-s + 371.·73-s + 367.·77-s − 1.26e3·79-s + 603.·83-s + ⋯ |
| L(s) = 1 | + 0.168·5-s + 0.377·7-s + 1.43·11-s + 0.0991·13-s − 1.40·17-s + 1.33·19-s + 0.764·23-s − 0.971·25-s + 0.722·29-s + 0.243·31-s + 0.0636·35-s − 1.74·37-s + 0.897·41-s + 0.299·43-s − 0.123·47-s + 0.142·49-s + 0.463·53-s + 0.242·55-s − 0.00507·59-s + 1.61·61-s + 0.0167·65-s + 0.408·67-s + 1.30·71-s + 0.596·73-s + 0.543·77-s − 1.80·79-s + 0.797·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.444745616\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.444745616\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - 7T \) |
| good | 5 | \( 1 - 1.88T + 125T^{2} \) |
| 11 | \( 1 - 52.4T + 1.33e3T^{2} \) |
| 13 | \( 1 - 4.64T + 2.19e3T^{2} \) |
| 17 | \( 1 + 98.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 110.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 84.2T + 1.21e4T^{2} \) |
| 29 | \( 1 - 112.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 42.0T + 2.97e4T^{2} \) |
| 37 | \( 1 + 392.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 235.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 84.4T + 7.95e4T^{2} \) |
| 47 | \( 1 + 39.8T + 1.03e5T^{2} \) |
| 53 | \( 1 - 178.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 2.29T + 2.05e5T^{2} \) |
| 61 | \( 1 - 768.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 224.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 780.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 371.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.26e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 603.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.18e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.59e3T + 9.12e5T^{2} \) |
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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
−9.812494027720295046186018899217, −9.106583215327333795144590389113, −8.390765050912754055150074148000, −7.19557481064945441946861328921, −6.55439949036628015412807541851, −5.47401168031004537485520483881, −4.45524614651661696936863329751, −3.49410572459355329603476829172, −2.09318995575273068944338599708, −0.931667795665787630048167871481,
0.931667795665787630048167871481, 2.09318995575273068944338599708, 3.49410572459355329603476829172, 4.45524614651661696936863329751, 5.47401168031004537485520483881, 6.55439949036628015412807541851, 7.19557481064945441946861328921, 8.390765050912754055150074148000, 9.106583215327333795144590389113, 9.812494027720295046186018899217
