| L(s) = 1 | + 2·2-s + 4·4-s − 5·5-s + 7·7-s + 8·8-s − 10·10-s − 12·11-s − 58·13-s + 14·14-s + 16·16-s − 42·17-s − 4·19-s − 20·20-s − 24·22-s − 24·23-s + 25·25-s − 116·26-s + 28·28-s − 294·29-s + 128·31-s + 32·32-s − 84·34-s − 35·35-s − 58·37-s − 8·38-s − 40·40-s − 282·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.377·7-s + 0.353·8-s − 0.316·10-s − 0.328·11-s − 1.23·13-s + 0.267·14-s + 1/4·16-s − 0.599·17-s − 0.0482·19-s − 0.223·20-s − 0.232·22-s − 0.217·23-s + 1/5·25-s − 0.874·26-s + 0.188·28-s − 1.88·29-s + 0.741·31-s + 0.176·32-s − 0.423·34-s − 0.169·35-s − 0.257·37-s − 0.0341·38-s − 0.158·40-s − 1.07·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - p T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + p T \) |
| 7 | \( 1 - p T \) |
| good | 11 | \( 1 + 12 T + p^{3} T^{2} \) |
| 13 | \( 1 + 58 T + p^{3} T^{2} \) |
| 17 | \( 1 + 42 T + p^{3} T^{2} \) |
| 19 | \( 1 + 4 T + p^{3} T^{2} \) |
| 23 | \( 1 + 24 T + p^{3} T^{2} \) |
| 29 | \( 1 + 294 T + p^{3} T^{2} \) |
| 31 | \( 1 - 128 T + p^{3} T^{2} \) |
| 37 | \( 1 + 58 T + p^{3} T^{2} \) |
| 41 | \( 1 + 282 T + p^{3} T^{2} \) |
| 43 | \( 1 - 428 T + p^{3} T^{2} \) |
| 47 | \( 1 + 384 T + p^{3} T^{2} \) |
| 53 | \( 1 - 138 T + p^{3} T^{2} \) |
| 59 | \( 1 + 468 T + p^{3} T^{2} \) |
| 61 | \( 1 + 250 T + p^{3} T^{2} \) |
| 67 | \( 1 + 556 T + p^{3} T^{2} \) |
| 71 | \( 1 + 624 T + p^{3} T^{2} \) |
| 73 | \( 1 + 958 T + p^{3} T^{2} \) |
| 79 | \( 1 - 8 p T + p^{3} T^{2} \) |
| 83 | \( 1 + 84 T + p^{3} T^{2} \) |
| 89 | \( 1 + 810 T + p^{3} T^{2} \) |
| 97 | \( 1 + 790 T + p^{3} T^{2} \) |
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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
−9.895358533704823070564077023432, −8.835397945131538246098263445118, −7.72609221902313261705354609505, −7.18105769615600731148195471561, −5.99029296703163388474459956736, −4.97951585256340814489389186169, −4.24523612742946532101079150027, −3.01373623048925687412957275797, −1.88284900626382667079589643737, 0,
1.88284900626382667079589643737, 3.01373623048925687412957275797, 4.24523612742946532101079150027, 4.97951585256340814489389186169, 5.99029296703163388474459956736, 7.18105769615600731148195471561, 7.72609221902313261705354609505, 8.835397945131538246098263445118, 9.895358533704823070564077023432
