| L(s) = 1 | + 7.53·3-s + 7.78·5-s + 5.94·7-s + 29.7·9-s + 21.2·11-s + 74.0·13-s + 58.6·15-s + 109.·17-s − 96.9·19-s + 44.7·21-s − 16.9·23-s − 64.4·25-s + 20.7·27-s + 29·29-s + 216.·31-s + 160.·33-s + 46.2·35-s + 138.·37-s + 557.·39-s + 344.·41-s − 61.6·43-s + 231.·45-s − 77.0·47-s − 307.·49-s + 821.·51-s − 19.8·53-s + 165.·55-s + ⋯ |
| L(s) = 1 | + 1.44·3-s + 0.696·5-s + 0.320·7-s + 1.10·9-s + 0.582·11-s + 1.58·13-s + 1.00·15-s + 1.55·17-s − 1.17·19-s + 0.465·21-s − 0.153·23-s − 0.515·25-s + 0.147·27-s + 0.185·29-s + 1.25·31-s + 0.844·33-s + 0.223·35-s + 0.615·37-s + 2.29·39-s + 1.31·41-s − 0.218·43-s + 0.767·45-s − 0.239·47-s − 0.897·49-s + 2.25·51-s − 0.0514·53-s + 0.405·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.621031330\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.621031330\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 29 | \( 1 - 29T \) |
| good | 3 | \( 1 - 7.53T + 27T^{2} \) |
| 5 | \( 1 - 7.78T + 125T^{2} \) |
| 7 | \( 1 - 5.94T + 343T^{2} \) |
| 11 | \( 1 - 21.2T + 1.33e3T^{2} \) |
| 13 | \( 1 - 74.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 109.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 96.9T + 6.85e3T^{2} \) |
| 23 | \( 1 + 16.9T + 1.21e4T^{2} \) |
| 31 | \( 1 - 216.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 138.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 344.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 61.6T + 7.95e4T^{2} \) |
| 47 | \( 1 + 77.0T + 1.03e5T^{2} \) |
| 53 | \( 1 + 19.8T + 1.48e5T^{2} \) |
| 59 | \( 1 + 732.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 310.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 765.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 74.3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 421.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.11e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.08e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 547.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 29.9T + 9.12e5T^{2} \) |
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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
−8.880799436507148609483449583629, −8.124100476276533413183111120793, −7.70451071033190385438833120152, −6.32935232097516079327979111120, −5.94818397281442969382979483075, −4.54594572780925327737187332711, −3.70549783708401849925375826876, −2.95031630101859559748233313322, −1.89110282452172163265027464947, −1.15580126746815094046533518946,
1.15580126746815094046533518946, 1.89110282452172163265027464947, 2.95031630101859559748233313322, 3.70549783708401849925375826876, 4.54594572780925327737187332711, 5.94818397281442969382979483075, 6.32935232097516079327979111120, 7.70451071033190385438833120152, 8.124100476276533413183111120793, 8.880799436507148609483449583629
