| L(s) = 1 | − 3·2-s + 4-s − 7·7-s + 21·8-s + 36·11-s + 34·13-s + 21·14-s − 71·16-s + 42·17-s − 124·19-s − 108·22-s − 102·26-s − 7·28-s − 102·29-s − 160·31-s + 45·32-s − 126·34-s − 398·37-s + 372·38-s + 318·41-s + 268·43-s + 36·44-s + 240·47-s + 49·49-s + 34·52-s − 498·53-s − 147·56-s + ⋯ |
| L(s) = 1 | − 1.06·2-s + 1/8·4-s − 0.377·7-s + 0.928·8-s + 0.986·11-s + 0.725·13-s + 0.400·14-s − 1.10·16-s + 0.599·17-s − 1.49·19-s − 1.04·22-s − 0.769·26-s − 0.0472·28-s − 0.653·29-s − 0.926·31-s + 0.248·32-s − 0.635·34-s − 1.76·37-s + 1.58·38-s + 1.21·41-s + 0.950·43-s + 0.123·44-s + 0.744·47-s + 1/7·49-s + 0.0906·52-s − 1.29·53-s − 0.350·56-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 | \( 1 + p T \) |
| good | 2 | \( 1 + 3 T + p^{3} T^{2} \) |
| 11 | \( 1 - 36 T + p^{3} T^{2} \) |
| 13 | \( 1 - 34 T + p^{3} T^{2} \) |
| 17 | \( 1 - 42 T + p^{3} T^{2} \) |
| 19 | \( 1 + 124 T + p^{3} T^{2} \) |
| 23 | \( 1 + p^{3} T^{2} \) |
| 29 | \( 1 + 102 T + p^{3} T^{2} \) |
| 31 | \( 1 + 160 T + p^{3} T^{2} \) |
| 37 | \( 1 + 398 T + p^{3} T^{2} \) |
| 41 | \( 1 - 318 T + p^{3} T^{2} \) |
| 43 | \( 1 - 268 T + p^{3} T^{2} \) |
| 47 | \( 1 - 240 T + p^{3} T^{2} \) |
| 53 | \( 1 + 498 T + p^{3} T^{2} \) |
| 59 | \( 1 - 132 T + p^{3} T^{2} \) |
| 61 | \( 1 - 398 T + p^{3} T^{2} \) |
| 67 | \( 1 + 92 T + p^{3} T^{2} \) |
| 71 | \( 1 - 720 T + p^{3} T^{2} \) |
| 73 | \( 1 - 502 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1024 T + p^{3} T^{2} \) |
| 83 | \( 1 + 204 T + p^{3} T^{2} \) |
| 89 | \( 1 + 354 T + p^{3} T^{2} \) |
| 97 | \( 1 - 286 T + p^{3} T^{2} \) |
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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
−8.885246046071588761565787055540, −8.063142983598651230061085054030, −7.19472202354178503994769808828, −6.44386508541565650041771898747, −5.51579502860220668657517786773, −4.25379457590558783859526953993, −3.62191593906425701898025457332, −2.07856042190099487141750439646, −1.12703829830999989009363196981, 0,
1.12703829830999989009363196981, 2.07856042190099487141750439646, 3.62191593906425701898025457332, 4.25379457590558783859526953993, 5.51579502860220668657517786773, 6.44386508541565650041771898747, 7.19472202354178503994769808828, 8.063142983598651230061085054030, 8.885246046071588761565787055540
