| L(s) = 1 | + 2-s + 4-s + 4.89·7-s + 8-s + 4.89·11-s − 2·13-s + 4.89·14-s + 16-s + 4.89·17-s + 4.89·22-s − 5·25-s − 2·26-s + 4.89·28-s + 6·29-s − 8·31-s + 32-s + 4.89·34-s + 4.89·37-s + 6·41-s − 9.79·43-s + 4.89·44-s + 16.9·49-s − 5·50-s − 2·52-s + 9.79·53-s + 4.89·56-s + 6·58-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 1.85·7-s + 0.353·8-s + 1.47·11-s − 0.554·13-s + 1.30·14-s + 0.250·16-s + 1.18·17-s + 1.04·22-s − 25-s − 0.392·26-s + 0.925·28-s + 1.11·29-s − 1.43·31-s + 0.176·32-s + 0.840·34-s + 0.805·37-s + 0.937·41-s − 1.49·43-s + 0.738·44-s + 2.42·49-s − 0.707·50-s − 0.277·52-s + 1.34·53-s + 0.654·56-s + 0.787·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9522 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9522 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.176394548\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.176394548\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 23 | \( 1 \) |
| good | 5 | \( 1 + 5T^{2} \) |
| 7 | \( 1 - 4.89T + 7T^{2} \) |
| 11 | \( 1 - 4.89T + 11T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 - 4.89T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 - 4.89T + 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 + 9.79T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 9.79T + 53T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 + 4.89T + 61T^{2} \) |
| 67 | \( 1 + 9.79T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 2T + 73T^{2} \) |
| 79 | \( 1 - 14.6T + 79T^{2} \) |
| 83 | \( 1 + 4.89T + 83T^{2} \) |
| 89 | \( 1 + 14.6T + 89T^{2} \) |
| 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
−7.58934386710987228702537535768, −7.09626816238028681111765748498, −6.18168449389959425260167492118, −5.49146855470918312953371974761, −4.95319099857161100736439600035, −4.17516906244859489806493090468, −3.73132860527077454920638907440, −2.57636997605370424880258527959, −1.70961937888180077446873151051, −1.10179826674082161989021170881,
1.10179826674082161989021170881, 1.70961937888180077446873151051, 2.57636997605370424880258527959, 3.73132860527077454920638907440, 4.17516906244859489806493090468, 4.95319099857161100736439600035, 5.49146855470918312953371974761, 6.18168449389959425260167492118, 7.09626816238028681111765748498, 7.58934386710987228702537535768
