| L(s) = 1 | + 2·2-s − 4·3-s + 4·4-s − 8·6-s − 7·7-s + 8·8-s − 11·9-s + 60·11-s − 16·12-s − 38·13-s − 14·14-s + 16·16-s − 42·17-s − 22·18-s − 52·19-s + 28·21-s + 120·22-s − 120·23-s − 32·24-s − 76·26-s + 152·27-s − 28·28-s − 234·29-s − 304·31-s + 32·32-s − 240·33-s − 84·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.769·3-s + 1/2·4-s − 0.544·6-s − 0.377·7-s + 0.353·8-s − 0.407·9-s + 1.64·11-s − 0.384·12-s − 0.810·13-s − 0.267·14-s + 1/4·16-s − 0.599·17-s − 0.288·18-s − 0.627·19-s + 0.290·21-s + 1.16·22-s − 1.08·23-s − 0.272·24-s − 0.573·26-s + 1.08·27-s − 0.188·28-s − 1.49·29-s − 1.76·31-s + 0.176·32-s − 1.26·33-s − 0.423·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - p T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + p T \) |
| good | 3 | \( 1 + 4 T + p^{3} T^{2} \) |
| 11 | \( 1 - 60 T + p^{3} T^{2} \) |
| 13 | \( 1 + 38 T + p^{3} T^{2} \) |
| 17 | \( 1 + 42 T + p^{3} T^{2} \) |
| 19 | \( 1 + 52 T + p^{3} T^{2} \) |
| 23 | \( 1 + 120 T + p^{3} T^{2} \) |
| 29 | \( 1 + 234 T + p^{3} T^{2} \) |
| 31 | \( 1 + 304 T + p^{3} T^{2} \) |
| 37 | \( 1 - 106 T + p^{3} T^{2} \) |
| 41 | \( 1 + 54 T + p^{3} T^{2} \) |
| 43 | \( 1 - 196 T + p^{3} T^{2} \) |
| 47 | \( 1 + 336 T + p^{3} T^{2} \) |
| 53 | \( 1 + 438 T + p^{3} T^{2} \) |
| 59 | \( 1 + 444 T + p^{3} T^{2} \) |
| 61 | \( 1 - 38 T + p^{3} T^{2} \) |
| 67 | \( 1 - 988 T + p^{3} T^{2} \) |
| 71 | \( 1 + 720 T + p^{3} T^{2} \) |
| 73 | \( 1 + 2 p T + p^{3} T^{2} \) |
| 79 | \( 1 + 808 T + p^{3} T^{2} \) |
| 83 | \( 1 + 612 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1146 T + p^{3} T^{2} \) |
| 97 | \( 1 - 70 T + p^{3} T^{2} \) |
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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
−11.01134479195513610671450819676, −9.750552157819167196193556983835, −8.859944742364187782911867263387, −7.39210064155521959827008538110, −6.42049730348402531031518873779, −5.79182796448254136711944431958, −4.56583699129886552541421384600, −3.55767192260810315828251616378, −1.93982667360229045690498247808, 0,
1.93982667360229045690498247808, 3.55767192260810315828251616378, 4.56583699129886552541421384600, 5.79182796448254136711944431958, 6.42049730348402531031518873779, 7.39210064155521959827008538110, 8.859944742364187782911867263387, 9.750552157819167196193556983835, 11.01134479195513610671450819676
