L(s) = 1 | + 2-s + 3-s + 4-s − 4·5-s + 6-s + 4·7-s + 8-s + 9-s − 4·10-s + 4·11-s + 12-s − 2·13-s + 4·14-s − 4·15-s + 16-s + 6·17-s + 18-s − 4·20-s + 4·21-s + 4·22-s − 6·23-s + 24-s + 11·25-s − 2·26-s + 27-s + 4·28-s − 4·29-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 1.78·5-s + 0.408·6-s + 1.51·7-s + 0.353·8-s + 1/3·9-s − 1.26·10-s + 1.20·11-s + 0.288·12-s − 0.554·13-s + 1.06·14-s − 1.03·15-s + 1/4·16-s + 1.45·17-s + 0.235·18-s − 0.894·20-s + 0.872·21-s + 0.852·22-s − 1.25·23-s + 0.204·24-s + 11/5·25-s − 0.392·26-s + 0.192·27-s + 0.755·28-s − 0.742·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 678 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 678 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.571760389\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.571760389\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 113 | \( 1 - T \) |
good | 5 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−10.88431765826710729196729988855, −9.572211897701689763969639801895, −8.423055390035103253206120384503, −7.67441940811279141783567526614, −7.43032595540450804018851718533, −5.88572827992595258362375659183, −4.48845006142529885757176593763, −4.17288707065704443058335825274, −3.08407932080882928721497987942, −1.44240333564762446333766353617,
1.44240333564762446333766353617, 3.08407932080882928721497987942, 4.17288707065704443058335825274, 4.48845006142529885757176593763, 5.88572827992595258362375659183, 7.43032595540450804018851718533, 7.67441940811279141783567526614, 8.423055390035103253206120384503, 9.572211897701689763969639801895, 10.88431765826710729196729988855