L(s) = 1 | + 2-s − 1.61·3-s + 4-s − 2.61·5-s − 1.61·6-s + 7-s + 8-s − 0.381·9-s − 2.61·10-s − 4.85·11-s − 1.61·12-s + 4.47·13-s + 14-s + 4.23·15-s + 16-s + 0.763·17-s − 0.381·18-s − 2.61·20-s − 1.61·21-s − 4.85·22-s + 8.94·23-s − 1.61·24-s + 1.85·25-s + 4.47·26-s + 5.47·27-s + 28-s − 0.145·29-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.934·3-s + 0.5·4-s − 1.17·5-s − 0.660·6-s + 0.377·7-s + 0.353·8-s − 0.127·9-s − 0.827·10-s − 1.46·11-s − 0.467·12-s + 1.24·13-s + 0.267·14-s + 1.09·15-s + 0.250·16-s + 0.185·17-s − 0.0900·18-s − 0.585·20-s − 0.353·21-s − 1.03·22-s + 1.86·23-s − 0.330·24-s + 0.370·25-s + 0.877·26-s + 1.05·27-s + 0.188·28-s − 0.0270·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 \) |
good | 3 | \( 1 + 1.61T + 3T^{2} \) |
| 5 | \( 1 + 2.61T + 5T^{2} \) |
| 11 | \( 1 + 4.85T + 11T^{2} \) |
| 13 | \( 1 - 4.47T + 13T^{2} \) |
| 17 | \( 1 - 0.763T + 17T^{2} \) |
| 23 | \( 1 - 8.94T + 23T^{2} \) |
| 29 | \( 1 + 0.145T + 29T^{2} \) |
| 31 | \( 1 + 6T + 31T^{2} \) |
| 37 | \( 1 + 7.85T + 37T^{2} \) |
| 41 | \( 1 - 9.56T + 41T^{2} \) |
| 43 | \( 1 - 3.85T + 43T^{2} \) |
| 47 | \( 1 + 10.8T + 47T^{2} \) |
| 53 | \( 1 - 5.14T + 53T^{2} \) |
| 59 | \( 1 - 11.5T + 59T^{2} \) |
| 61 | \( 1 + 8.56T + 61T^{2} \) |
| 67 | \( 1 + 5.23T + 67T^{2} \) |
| 71 | \( 1 + 8.56T + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 + 0.326T + 79T^{2} \) |
| 83 | \( 1 - 11.2T + 83T^{2} \) |
| 89 | \( 1 - 3.14T + 89T^{2} \) |
| 97 | \( 1 + 7.14T + 97T^{2} \) |
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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
−7.72872323657500873769468551289, −7.12740480700960409061843157223, −6.30264129960489709968929440561, −5.38602054531856021039119490519, −5.17483758394118768884653336537, −4.22529966535890295866177387005, −3.43331886074670664430295809292, −2.67371383952983238738960783381, −1.19788232621537924725881108099, 0,
1.19788232621537924725881108099, 2.67371383952983238738960783381, 3.43331886074670664430295809292, 4.22529966535890295866177387005, 5.17483758394118768884653336537, 5.38602054531856021039119490519, 6.30264129960489709968929440561, 7.12740480700960409061843157223, 7.72872323657500873769468551289