| L(s) = 1 | − 2-s + 4-s − 3·5-s − 8-s + 3·10-s + 6·11-s + 5·13-s + 16-s + 2·19-s − 3·20-s − 6·22-s + 8·23-s + 4·25-s − 5·26-s − 29-s − 5·31-s − 32-s + 2·37-s − 2·38-s + 3·40-s + 5·41-s + 4·43-s + 6·44-s − 8·46-s + 5·47-s − 4·50-s + 5·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 1.34·5-s − 0.353·8-s + 0.948·10-s + 1.80·11-s + 1.38·13-s + 1/4·16-s + 0.458·19-s − 0.670·20-s − 1.27·22-s + 1.66·23-s + 4/5·25-s − 0.980·26-s − 0.185·29-s − 0.898·31-s − 0.176·32-s + 0.328·37-s − 0.324·38-s + 0.474·40-s + 0.780·41-s + 0.609·43-s + 0.904·44-s − 1.17·46-s + 0.729·47-s − 0.565·50-s + 0.693·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 254898 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 254898 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 5 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 + 15 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 13 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−12.86413743557425, −12.53441034083152, −11.85252944740573, −11.69980969200978, −11.16234747018991, −10.89280720584991, −10.49135804282762, −9.589823043567735, −9.182400436792407, −8.903565524140808, −8.549727146218678, −7.916801287233513, −7.353745902819516, −7.140373445012024, −6.614538070222561, −5.914566610961740, −5.712498180921219, −4.672728365836503, −4.200040580836945, −3.851452468741511, −3.226577334632762, −2.912403568075518, −1.818915610170119, −1.166585084628537, −0.9463324567409532, 0,
0.9463324567409532, 1.166585084628537, 1.818915610170119, 2.912403568075518, 3.226577334632762, 3.851452468741511, 4.200040580836945, 4.672728365836503, 5.712498180921219, 5.914566610961740, 6.614538070222561, 7.140373445012024, 7.353745902819516, 7.916801287233513, 8.549727146218678, 8.903565524140808, 9.182400436792407, 9.589823043567735, 10.49135804282762, 10.89280720584991, 11.16234747018991, 11.69980969200978, 11.85252944740573, 12.53441034083152, 12.86413743557425
