L(s) = 1 | + 2-s + 2.85·3-s + 4-s + 0.532·5-s + 2.85·6-s + 7-s + 8-s + 5.14·9-s + 0.532·10-s + 2.71·11-s + 2.85·12-s + 14-s + 1.51·15-s + 16-s − 2.94·17-s + 5.14·18-s + 5.83·19-s + 0.532·20-s + 2.85·21-s + 2.71·22-s − 5.52·23-s + 2.85·24-s − 4.71·25-s + 6.11·27-s + 28-s − 8.19·29-s + 1.51·30-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.64·3-s + 0.5·4-s + 0.237·5-s + 1.16·6-s + 0.377·7-s + 0.353·8-s + 1.71·9-s + 0.168·10-s + 0.818·11-s + 0.823·12-s + 0.267·14-s + 0.392·15-s + 0.250·16-s − 0.713·17-s + 1.21·18-s + 1.33·19-s + 0.118·20-s + 0.622·21-s + 0.578·22-s − 1.15·23-s + 0.582·24-s − 0.943·25-s + 1.17·27-s + 0.188·28-s − 1.52·29-s + 0.277·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.645552937\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.645552937\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 3 | \( 1 - 2.85T + 3T^{2} \) |
| 5 | \( 1 - 0.532T + 5T^{2} \) |
| 11 | \( 1 - 2.71T + 11T^{2} \) |
| 17 | \( 1 + 2.94T + 17T^{2} \) |
| 19 | \( 1 - 5.83T + 19T^{2} \) |
| 23 | \( 1 + 5.52T + 23T^{2} \) |
| 29 | \( 1 + 8.19T + 29T^{2} \) |
| 31 | \( 1 - 3.42T + 31T^{2} \) |
| 37 | \( 1 - 6.00T + 37T^{2} \) |
| 41 | \( 1 + 7.76T + 41T^{2} \) |
| 43 | \( 1 + 9.13T + 43T^{2} \) |
| 47 | \( 1 + 9.98T + 47T^{2} \) |
| 53 | \( 1 - 12.8T + 53T^{2} \) |
| 59 | \( 1 + 1.24T + 59T^{2} \) |
| 61 | \( 1 - 1.36T + 61T^{2} \) |
| 67 | \( 1 + 9.49T + 67T^{2} \) |
| 71 | \( 1 - 1.34T + 71T^{2} \) |
| 73 | \( 1 - 11.2T + 73T^{2} \) |
| 79 | \( 1 - 7.62T + 79T^{2} \) |
| 83 | \( 1 + 3.49T + 83T^{2} \) |
| 89 | \( 1 + 14.5T + 89T^{2} \) |
| 97 | \( 1 - 7.63T + 97T^{2} \) |
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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
−8.925843510482283031480323060294, −8.147091295575915315188973309302, −7.56387565620167676926935509875, −6.75666615639916406994883872858, −5.82607560415971956864905926088, −4.78179818682738036376226575287, −3.87168810688954574872618716492, −3.36829189168726227403589618691, −2.24398704396652079212291353105, −1.60836176932259246917542507243,
1.60836176932259246917542507243, 2.24398704396652079212291353105, 3.36829189168726227403589618691, 3.87168810688954574872618716492, 4.78179818682738036376226575287, 5.82607560415971956864905926088, 6.75666615639916406994883872858, 7.56387565620167676926935509875, 8.147091295575915315188973309302, 8.925843510482283031480323060294