L(s) = 1 | + 2-s + 4-s + 2.34·5-s − 1.28·7-s + 8-s + 2.34·10-s − 5.75·11-s − 0.304·13-s − 1.28·14-s + 16-s − 4.18·17-s + 2.34·20-s − 5.75·22-s + 6.47·23-s + 0.497·25-s − 0.304·26-s − 1.28·28-s − 3.12·29-s − 6.44·31-s + 32-s − 4.18·34-s − 3.01·35-s + 3.97·37-s + 2.34·40-s − 5.01·41-s − 0.989·43-s − 5.75·44-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.5·4-s + 1.04·5-s − 0.485·7-s + 0.353·8-s + 0.741·10-s − 1.73·11-s − 0.0843·13-s − 0.343·14-s + 0.250·16-s − 1.01·17-s + 0.524·20-s − 1.22·22-s + 1.35·23-s + 0.0994·25-s − 0.0596·26-s − 0.242·28-s − 0.580·29-s − 1.15·31-s + 0.176·32-s − 0.717·34-s − 0.508·35-s + 0.654·37-s + 0.370·40-s − 0.783·41-s − 0.150·43-s − 0.867·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6498 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6498 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 \) |
| 19 | \( 1 \) |
good | 5 | \( 1 - 2.34T + 5T^{2} \) |
| 7 | \( 1 + 1.28T + 7T^{2} \) |
| 11 | \( 1 + 5.75T + 11T^{2} \) |
| 13 | \( 1 + 0.304T + 13T^{2} \) |
| 17 | \( 1 + 4.18T + 17T^{2} \) |
| 23 | \( 1 - 6.47T + 23T^{2} \) |
| 29 | \( 1 + 3.12T + 29T^{2} \) |
| 31 | \( 1 + 6.44T + 31T^{2} \) |
| 37 | \( 1 - 3.97T + 37T^{2} \) |
| 41 | \( 1 + 5.01T + 41T^{2} \) |
| 43 | \( 1 + 0.989T + 43T^{2} \) |
| 47 | \( 1 - 4.39T + 47T^{2} \) |
| 53 | \( 1 - 3.29T + 53T^{2} \) |
| 59 | \( 1 - 3.31T + 59T^{2} \) |
| 61 | \( 1 + 10.9T + 61T^{2} \) |
| 67 | \( 1 - 4.38T + 67T^{2} \) |
| 71 | \( 1 + 4.41T + 71T^{2} \) |
| 73 | \( 1 + 2.26T + 73T^{2} \) |
| 79 | \( 1 + 8.57T + 79T^{2} \) |
| 83 | \( 1 - 9.76T + 83T^{2} \) |
| 89 | \( 1 + 15.0T + 89T^{2} \) |
| 97 | \( 1 + 17.3T + 97T^{2} \) |
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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
−7.43818020376071750942532793548, −6.87541629228144831782447653635, −6.09482418143873260348988176613, −5.42748147601726906331582804849, −5.03039145691331240965360041576, −4.06581922281404590057284784195, −2.99071172836294244258984442077, −2.51195860906334722378037569372, −1.63896316358874672567725334531, 0,
1.63896316358874672567725334531, 2.51195860906334722378037569372, 2.99071172836294244258984442077, 4.06581922281404590057284784195, 5.03039145691331240965360041576, 5.42748147601726906331582804849, 6.09482418143873260348988176613, 6.87541629228144831782447653635, 7.43818020376071750942532793548