| L(s) = 1 | + 2·2-s + 4·4-s + 15.9·5-s + 7·7-s + 8·8-s + 31.8·10-s + 2.05·11-s + 79.6·13-s + 14·14-s + 16·16-s − 67.5·17-s − 108.·19-s + 63.7·20-s + 4.10·22-s − 7.74·23-s + 129.·25-s + 159.·26-s + 28·28-s + 249.·29-s + 5.30·31-s + 32·32-s − 135.·34-s + 111.·35-s − 48.6·37-s − 216.·38-s + 127.·40-s − 45.0·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 1.42·5-s + 0.377·7-s + 0.353·8-s + 1.00·10-s + 0.0561·11-s + 1.70·13-s + 0.267·14-s + 0.250·16-s − 0.964·17-s − 1.30·19-s + 0.713·20-s + 0.0397·22-s − 0.0702·23-s + 1.03·25-s + 1.20·26-s + 0.188·28-s + 1.59·29-s + 0.0307·31-s + 0.176·32-s − 0.681·34-s + 0.539·35-s − 0.216·37-s − 0.925·38-s + 0.504·40-s − 0.171·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.102906665\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.102906665\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - 7T \) |
| good | 5 | \( 1 - 15.9T + 125T^{2} \) |
| 11 | \( 1 - 2.05T + 1.33e3T^{2} \) |
| 13 | \( 1 - 79.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 67.5T + 4.91e3T^{2} \) |
| 19 | \( 1 + 108.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 7.74T + 1.21e4T^{2} \) |
| 29 | \( 1 - 249.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 5.30T + 2.97e4T^{2} \) |
| 37 | \( 1 + 48.6T + 5.06e4T^{2} \) |
| 41 | \( 1 + 45.0T + 6.89e4T^{2} \) |
| 43 | \( 1 + 201.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 4.49T + 1.03e5T^{2} \) |
| 53 | \( 1 - 279.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 675.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 842.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 329.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 811.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 635.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 480.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.00e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.37e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.67e3T + 9.12e5T^{2} \) |
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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
−10.83159264280909969550259373526, −10.33236311639254333456613256461, −9.020046917675173344337172567260, −8.308856684807674487784012744881, −6.59084361426377777105608391437, −6.21536901260908837990770594702, −5.11014234247036368588082958837, −3.99910629239573571581173260872, −2.49070214474638011135086785958, −1.44460749967432852230676224822,
1.44460749967432852230676224822, 2.49070214474638011135086785958, 3.99910629239573571581173260872, 5.11014234247036368588082958837, 6.21536901260908837990770594702, 6.59084361426377777105608391437, 8.308856684807674487784012744881, 9.020046917675173344337172567260, 10.33236311639254333456613256461, 10.83159264280909969550259373526
