| L(s) = 1 | − 9.75·3-s − 18.3·5-s − 34.4·7-s + 68.1·9-s + 11.0·11-s + 57.7·13-s + 179.·15-s − 53.0·17-s + 93.5·19-s + 335.·21-s − 61.2·23-s + 213.·25-s − 401.·27-s + 154.·29-s − 169.·31-s − 108.·33-s + 633.·35-s + 37·37-s − 563.·39-s − 162.·41-s + 134.·43-s − 1.25e3·45-s − 359.·47-s + 841.·49-s + 517.·51-s − 192.·53-s − 204.·55-s + ⋯ |
| L(s) = 1 | − 1.87·3-s − 1.64·5-s − 1.85·7-s + 2.52·9-s + 0.304·11-s + 1.23·13-s + 3.08·15-s − 0.756·17-s + 1.13·19-s + 3.48·21-s − 0.555·23-s + 1.70·25-s − 2.86·27-s + 0.992·29-s − 0.980·31-s − 0.571·33-s + 3.05·35-s + 0.164·37-s − 2.31·39-s − 0.620·41-s + 0.477·43-s − 4.15·45-s − 1.11·47-s + 2.45·49-s + 1.42·51-s − 0.499·53-s − 0.500·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 37 | \( 1 - 37T \) |
| good | 3 | \( 1 + 9.75T + 27T^{2} \) |
| 5 | \( 1 + 18.3T + 125T^{2} \) |
| 7 | \( 1 + 34.4T + 343T^{2} \) |
| 11 | \( 1 - 11.0T + 1.33e3T^{2} \) |
| 13 | \( 1 - 57.7T + 2.19e3T^{2} \) |
| 17 | \( 1 + 53.0T + 4.91e3T^{2} \) |
| 19 | \( 1 - 93.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + 61.2T + 1.21e4T^{2} \) |
| 29 | \( 1 - 154.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 169.T + 2.97e4T^{2} \) |
| 41 | \( 1 + 162.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 134.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 359.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 192.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 472.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 1.07T + 2.26e5T^{2} \) |
| 67 | \( 1 - 56.3T + 3.00e5T^{2} \) |
| 71 | \( 1 - 607.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 237.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.18e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 969.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 197.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.05e3T + 9.12e5T^{2} \) |
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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
−10.10400445600178484898186537257, −9.141866487755550102103200679644, −7.80244214234859842139807575453, −6.69285865263574423075329823303, −6.47493261016388181675981439181, −5.29452380480090722313261748404, −4.06969881681793352947489764641, −3.47563632663080182107630923454, −0.849131449637972088326381768929, 0,
0.849131449637972088326381768929, 3.47563632663080182107630923454, 4.06969881681793352947489764641, 5.29452380480090722313261748404, 6.47493261016388181675981439181, 6.69285865263574423075329823303, 7.80244214234859842139807575453, 9.141866487755550102103200679644, 10.10400445600178484898186537257
