| L(s) = 1 | − 2-s − 3.08·3-s + 4-s + 5-s + 3.08·6-s − 0.803·7-s − 8-s + 6.51·9-s − 10-s + 1.13·11-s − 3.08·12-s − 13-s + 0.803·14-s − 3.08·15-s + 16-s − 3.74·17-s − 6.51·18-s + 5.76·19-s + 20-s + 2.47·21-s − 1.13·22-s − 0.527·23-s + 3.08·24-s + 25-s + 26-s − 10.8·27-s − 0.803·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.78·3-s + 0.5·4-s + 0.447·5-s + 1.25·6-s − 0.303·7-s − 0.353·8-s + 2.17·9-s − 0.316·10-s + 0.341·11-s − 0.890·12-s − 0.277·13-s + 0.214·14-s − 0.796·15-s + 0.250·16-s − 0.907·17-s − 1.53·18-s + 1.32·19-s + 0.223·20-s + 0.541·21-s − 0.241·22-s − 0.109·23-s + 0.629·24-s + 0.200·25-s + 0.196·26-s − 2.08·27-s − 0.151·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| good | 3 | \( 1 + 3.08T + 3T^{2} \) |
| 7 | \( 1 + 0.803T + 7T^{2} \) |
| 11 | \( 1 - 1.13T + 11T^{2} \) |
| 17 | \( 1 + 3.74T + 17T^{2} \) |
| 19 | \( 1 - 5.76T + 19T^{2} \) |
| 23 | \( 1 + 0.527T + 23T^{2} \) |
| 29 | \( 1 + 6.04T + 29T^{2} \) |
| 37 | \( 1 - 3.28T + 37T^{2} \) |
| 41 | \( 1 + 4.91T + 41T^{2} \) |
| 43 | \( 1 + 4.43T + 43T^{2} \) |
| 47 | \( 1 - 5.06T + 47T^{2} \) |
| 53 | \( 1 + 8.77T + 53T^{2} \) |
| 59 | \( 1 - 8.40T + 59T^{2} \) |
| 61 | \( 1 + 0.528T + 61T^{2} \) |
| 67 | \( 1 - 3.70T + 67T^{2} \) |
| 71 | \( 1 - 6.17T + 71T^{2} \) |
| 73 | \( 1 - 3.50T + 73T^{2} \) |
| 79 | \( 1 + 7.43T + 79T^{2} \) |
| 83 | \( 1 + 4.62T + 83T^{2} \) |
| 89 | \( 1 - 4.42T + 89T^{2} \) |
| 97 | \( 1 - 9.35T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.940987064180950383373628914364, −7.08275264406870118956538874613, −6.64395319687452081472746568973, −5.92955516782975215646837771214, −5.30767703463609375159774424309, −4.55468710181897756392780922628, −3.42541302542875636749421159503, −2.05559278507478746128945949886, −1.08179510647468563147882039993, 0,
1.08179510647468563147882039993, 2.05559278507478746128945949886, 3.42541302542875636749421159503, 4.55468710181897756392780922628, 5.30767703463609375159774424309, 5.92955516782975215646837771214, 6.64395319687452081472746568973, 7.08275264406870118956538874613, 7.940987064180950383373628914364
