| L(s) = 1 | − 2-s + 4-s − 5-s − 8-s + 10-s + 16-s + 17-s + 2·19-s − 20-s + 25-s + 8·29-s − 8·31-s − 32-s − 34-s + 6·37-s − 2·38-s + 40-s − 6·41-s − 4·43-s − 7·49-s − 50-s + 12·53-s − 8·58-s − 2·59-s − 8·61-s + 8·62-s + 64-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.353·8-s + 0.316·10-s + 1/4·16-s + 0.242·17-s + 0.458·19-s − 0.223·20-s + 1/5·25-s + 1.48·29-s − 1.43·31-s − 0.176·32-s − 0.171·34-s + 0.986·37-s − 0.324·38-s + 0.158·40-s − 0.937·41-s − 0.609·43-s − 49-s − 0.141·50-s + 1.64·53-s − 1.05·58-s − 0.260·59-s − 1.02·61-s + 1.01·62-s + 1/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 258570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 258570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.525410037\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.525410037\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−12.76806396094665, −12.31845279511595, −11.64826754809345, −11.53595193311367, −11.01742192047767, −10.33534156050523, −10.11545644366319, −9.635346356168681, −8.963295548507972, −8.698388529553451, −8.186170828371194, −7.609675156062309, −7.377748303014305, −6.720617505241656, −6.323179948498582, −5.745350931276425, −5.111558167734402, −4.710118050567181, −4.001747872057167, −3.390379445602651, −3.041237207039903, −2.280049103986347, −1.724132061702402, −0.9876841014042345, −0.4416785695900186,
0.4416785695900186, 0.9876841014042345, 1.724132061702402, 2.280049103986347, 3.041237207039903, 3.390379445602651, 4.001747872057167, 4.710118050567181, 5.111558167734402, 5.745350931276425, 6.323179948498582, 6.720617505241656, 7.377748303014305, 7.609675156062309, 8.186170828371194, 8.698388529553451, 8.963295548507972, 9.635346356168681, 10.11545644366319, 10.33534156050523, 11.01742192047767, 11.53595193311367, 11.64826754809345, 12.31845279511595, 12.76806396094665
