| L(s) = 1 | − 4.64·3-s − 12.8·5-s + 26.0·7-s − 5.41·9-s + 62.8·11-s − 22.3·13-s + 59.8·15-s − 57.9·17-s − 71.3·19-s − 121.·21-s − 49.5·23-s + 40.7·25-s + 150.·27-s + 29·29-s + 62.9·31-s − 291.·33-s − 335.·35-s − 119.·37-s + 104.·39-s − 414.·41-s + 348.·43-s + 69.7·45-s + 553.·47-s + 335.·49-s + 269.·51-s + 107.·53-s − 808.·55-s + ⋯ |
| L(s) = 1 | − 0.894·3-s − 1.15·5-s + 1.40·7-s − 0.200·9-s + 1.72·11-s − 0.477·13-s + 1.02·15-s − 0.827·17-s − 0.861·19-s − 1.25·21-s − 0.449·23-s + 0.325·25-s + 1.07·27-s + 0.185·29-s + 0.364·31-s − 1.53·33-s − 1.61·35-s − 0.529·37-s + 0.427·39-s − 1.58·41-s + 1.23·43-s + 0.231·45-s + 1.71·47-s + 0.979·49-s + 0.739·51-s + 0.278·53-s − 1.98·55-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
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 29 | \( 1 - 29T \) |
| good | 3 | \( 1 + 4.64T + 27T^{2} \) |
| 5 | \( 1 + 12.8T + 125T^{2} \) |
| 7 | \( 1 - 26.0T + 343T^{2} \) |
| 11 | \( 1 - 62.8T + 1.33e3T^{2} \) |
| 13 | \( 1 + 22.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 57.9T + 4.91e3T^{2} \) |
| 19 | \( 1 + 71.3T + 6.85e3T^{2} \) |
| 23 | \( 1 + 49.5T + 1.21e4T^{2} \) |
| 31 | \( 1 - 62.9T + 2.97e4T^{2} \) |
| 37 | \( 1 + 119.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 414.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 348.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 553.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 107.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 136.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 579.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 919.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 781.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 133.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 868.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 83.3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 357.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 187.T + 9.12e5T^{2} \) |
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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
−8.549531680709104050566534409993, −7.68930020732539744120827903688, −6.86184905024920614343300702064, −6.15478506755053183201375239229, −5.08996763030716258396747875615, −4.38095073200934793466793782573, −3.81267031600309620110055640095, −2.22384003129698242774997197889, −1.07503721291844775694128717517, 0,
1.07503721291844775694128717517, 2.22384003129698242774997197889, 3.81267031600309620110055640095, 4.38095073200934793466793782573, 5.08996763030716258396747875615, 6.15478506755053183201375239229, 6.86184905024920614343300702064, 7.68930020732539744120827903688, 8.549531680709104050566534409993
