| L(s) = 1 | − 2-s + 4-s − 5-s + 7-s − 8-s − 3·9-s + 10-s + 4·11-s − 13-s − 14-s + 16-s + 2·17-s + 3·18-s − 4·19-s − 20-s − 4·22-s + 8·23-s + 25-s + 26-s + 28-s − 2·29-s + 4·31-s − 32-s − 2·34-s − 35-s − 3·36-s + 10·37-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.377·7-s − 0.353·8-s − 9-s + 0.316·10-s + 1.20·11-s − 0.277·13-s − 0.267·14-s + 1/4·16-s + 0.485·17-s + 0.707·18-s − 0.917·19-s − 0.223·20-s − 0.852·22-s + 1.66·23-s + 1/5·25-s + 0.196·26-s + 0.188·28-s − 0.371·29-s + 0.718·31-s − 0.176·32-s − 0.342·34-s − 0.169·35-s − 1/2·36-s + 1.64·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 910 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 910 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.050905834\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.050905834\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−10.03401927531261292069804759359, −9.023484582452833896797067036074, −8.606634794844299052227542149321, −7.68443512123396537781239993732, −6.80887686277379917162235182824, −5.94625299246196711614816800877, −4.78804965180790131377979670157, −3.61086575373742178283982348040, −2.46216988198852531957525169063, −0.926485716749688229635470652844,
0.926485716749688229635470652844, 2.46216988198852531957525169063, 3.61086575373742178283982348040, 4.78804965180790131377979670157, 5.94625299246196711614816800877, 6.80887686277379917162235182824, 7.68443512123396537781239993732, 8.606634794844299052227542149321, 9.023484582452833896797067036074, 10.03401927531261292069804759359
