| L(s) = 1 | + 2-s + 4-s − 2.95·5-s + 3.20·7-s + 8-s − 2.95·10-s − 2.20·11-s + 6.95·13-s + 3.20·14-s + 16-s + 3.55·17-s + 4.75·19-s − 2.95·20-s − 2.20·22-s − 5.71·23-s + 3.75·25-s + 6.95·26-s + 3.20·28-s + 2.79·29-s − 4.35·31-s + 32-s + 3.55·34-s − 9.47·35-s − 11.6·37-s + 4.75·38-s − 2.95·40-s − 4.95·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 1.32·5-s + 1.21·7-s + 0.353·8-s − 0.935·10-s − 0.663·11-s + 1.93·13-s + 0.855·14-s + 0.250·16-s + 0.862·17-s + 1.09·19-s − 0.661·20-s − 0.469·22-s − 1.19·23-s + 0.751·25-s + 1.36·26-s + 0.605·28-s + 0.519·29-s − 0.782·31-s + 0.176·32-s + 0.609·34-s − 1.60·35-s − 1.91·37-s + 0.771·38-s − 0.467·40-s − 0.773·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.997860735\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.997860735\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 223 | \( 1 - T \) |
| good | 5 | \( 1 + 2.95T + 5T^{2} \) |
| 7 | \( 1 - 3.20T + 7T^{2} \) |
| 11 | \( 1 + 2.20T + 11T^{2} \) |
| 13 | \( 1 - 6.95T + 13T^{2} \) |
| 17 | \( 1 - 3.55T + 17T^{2} \) |
| 19 | \( 1 - 4.75T + 19T^{2} \) |
| 23 | \( 1 + 5.71T + 23T^{2} \) |
| 29 | \( 1 - 2.79T + 29T^{2} \) |
| 31 | \( 1 + 4.35T + 31T^{2} \) |
| 37 | \( 1 + 11.6T + 37T^{2} \) |
| 41 | \( 1 + 4.95T + 41T^{2} \) |
| 43 | \( 1 - 3.35T + 43T^{2} \) |
| 47 | \( 1 - 7.36T + 47T^{2} \) |
| 53 | \( 1 - 8.67T + 53T^{2} \) |
| 59 | \( 1 - 11.8T + 59T^{2} \) |
| 61 | \( 1 + 5.91T + 61T^{2} \) |
| 67 | \( 1 + 4.87T + 67T^{2} \) |
| 71 | \( 1 - 4.59T + 71T^{2} \) |
| 73 | \( 1 - 0.749T + 73T^{2} \) |
| 79 | \( 1 - 12.9T + 79T^{2} \) |
| 83 | \( 1 + 15.4T + 83T^{2} \) |
| 89 | \( 1 - 18.5T + 89T^{2} \) |
| 97 | \( 1 - 1.95T + 97T^{2} \) |
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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
−8.341871777162703935825304404849, −7.64712102595978213019073337404, −7.21241186624343474639211307303, −6.00339568473177596723570548557, −5.38911737832299879927137913686, −4.64542513836203150636448811299, −3.64616715727597076811049171137, −3.49162162842925893987519108614, −1.99276696267867831427129386799, −0.941916445829507606700782102901,
0.941916445829507606700782102901, 1.99276696267867831427129386799, 3.49162162842925893987519108614, 3.64616715727597076811049171137, 4.64542513836203150636448811299, 5.38911737832299879927137913686, 6.00339568473177596723570548557, 7.21241186624343474639211307303, 7.64712102595978213019073337404, 8.341871777162703935825304404849
