L(s) = 1 | + 2.35·2-s + 3·3-s − 2.47·4-s + 17.9·5-s + 7.05·6-s + 10.1·7-s − 24.6·8-s + 9·9-s + 42.2·10-s − 11·11-s − 7.41·12-s − 88.1·13-s + 23.7·14-s + 53.8·15-s − 38.1·16-s − 13.2·17-s + 21.1·18-s − 38.0·19-s − 44.3·20-s + 30.3·21-s − 25.8·22-s − 123.·23-s − 73.8·24-s + 197.·25-s − 207.·26-s + 27·27-s − 25.0·28-s + ⋯ |
L(s) = 1 | + 0.831·2-s + 0.577·3-s − 0.308·4-s + 1.60·5-s + 0.479·6-s + 0.546·7-s − 1.08·8-s + 0.333·9-s + 1.33·10-s − 0.301·11-s − 0.178·12-s − 1.88·13-s + 0.454·14-s + 0.927·15-s − 0.595·16-s − 0.189·17-s + 0.277·18-s − 0.459·19-s − 0.496·20-s + 0.315·21-s − 0.250·22-s − 1.12·23-s − 0.628·24-s + 1.58·25-s − 1.56·26-s + 0.192·27-s − 0.168·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - 3T \) |
| 11 | \( 1 + 11T \) |
| 61 | \( 1 + 61T \) |
good | 2 | \( 1 - 2.35T + 8T^{2} \) |
| 5 | \( 1 - 17.9T + 125T^{2} \) |
| 7 | \( 1 - 10.1T + 343T^{2} \) |
| 13 | \( 1 + 88.1T + 2.19e3T^{2} \) |
| 17 | \( 1 + 13.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 38.0T + 6.85e3T^{2} \) |
| 23 | \( 1 + 123.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 227.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 259.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 11.7T + 5.06e4T^{2} \) |
| 41 | \( 1 - 126.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 497.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 342.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 143.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 481.T + 2.05e5T^{2} \) |
| 67 | \( 1 + 703.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 156.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 878.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 296.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 801.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.26e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 528.T + 9.12e5T^{2} \) |
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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
−8.517232095307051863233368665065, −7.64117360657108697154372103598, −6.58318061911926075864745903298, −5.87492497159211568407338850574, −4.88668406766589068234765048307, −4.71383104555720372284252031565, −3.29538009466649228468480475687, −2.40269345534345557051419290284, −1.77264320166331055913173389941, 0,
1.77264320166331055913173389941, 2.40269345534345557051419290284, 3.29538009466649228468480475687, 4.71383104555720372284252031565, 4.88668406766589068234765048307, 5.87492497159211568407338850574, 6.58318061911926075864745903298, 7.64117360657108697154372103598, 8.517232095307051863233368665065