| L(s) = 1 | + 0.428·2-s − 7.81·4-s + 7·7-s − 6.77·8-s + 27.4·11-s − 46.5·13-s + 2.99·14-s + 59.6·16-s − 5.20·17-s − 91.0·19-s + 11.7·22-s + 111.·23-s − 19.9·26-s − 54.7·28-s − 0.0763·29-s + 201.·31-s + 79.7·32-s − 2.22·34-s − 312.·37-s − 38.9·38-s − 102.·41-s + 257.·43-s − 214.·44-s + 47.7·46-s + 350.·47-s + 49·49-s + 363.·52-s + ⋯ |
| L(s) = 1 | + 0.151·2-s − 0.977·4-s + 0.377·7-s − 0.299·8-s + 0.753·11-s − 0.993·13-s + 0.0572·14-s + 0.931·16-s − 0.0742·17-s − 1.09·19-s + 0.114·22-s + 1.01·23-s − 0.150·26-s − 0.369·28-s − 0.000488·29-s + 1.16·31-s + 0.440·32-s − 0.0112·34-s − 1.39·37-s − 0.166·38-s − 0.390·41-s + 0.912·43-s − 0.735·44-s + 0.153·46-s + 1.08·47-s + 0.142·49-s + 0.970·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - 7T \) |
| good | 2 | \( 1 - 0.428T + 8T^{2} \) |
| 11 | \( 1 - 27.4T + 1.33e3T^{2} \) |
| 13 | \( 1 + 46.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 5.20T + 4.91e3T^{2} \) |
| 19 | \( 1 + 91.0T + 6.85e3T^{2} \) |
| 23 | \( 1 - 111.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 0.0763T + 2.43e4T^{2} \) |
| 31 | \( 1 - 201.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 312.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 102.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 257.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 350.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 196.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 881.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 737.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 365.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.11e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 261.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 273.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 87.1T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.09e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 228.T + 9.12e5T^{2} \) |
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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
−8.802161211045283247328041556638, −7.986686370608416762180177822359, −7.05656419249927062493227498527, −6.17486841702477602334989389582, −5.11823613710478722651983582791, −4.55067836686784894620012984424, −3.70344662651772696161070364345, −2.52732059021435602880065080430, −1.20128924277577842008464455453, 0,
1.20128924277577842008464455453, 2.52732059021435602880065080430, 3.70344662651772696161070364345, 4.55067836686784894620012984424, 5.11823613710478722651983582791, 6.17486841702477602334989389582, 7.05656419249927062493227498527, 7.986686370608416762180177822359, 8.802161211045283247328041556638
