| L(s) = 1 | − 2-s − 3-s + 4-s + 2·5-s + 6-s − 8-s + 9-s − 2·10-s + 2·11-s − 12-s + 4·13-s − 2·15-s + 16-s + 2·17-s − 18-s + 19-s + 2·20-s − 2·22-s − 2·23-s + 24-s − 25-s − 4·26-s − 27-s + 2·29-s + 2·30-s + 2·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.894·5-s + 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.632·10-s + 0.603·11-s − 0.288·12-s + 1.10·13-s − 0.516·15-s + 1/4·16-s + 0.485·17-s − 0.235·18-s + 0.229·19-s + 0.447·20-s − 0.426·22-s − 0.417·23-s + 0.204·24-s − 1/5·25-s − 0.784·26-s − 0.192·27-s + 0.371·29-s + 0.365·30-s + 0.359·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.663288160\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.663288160\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| 53 | \( 1 + T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 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
−8.055208100493187914187647428019, −7.42529020217861739369672005166, −6.49129224933206391529532127711, −6.04753499543911157371423859918, −5.56427106329783167101003842954, −4.45755962899748770496251019839, −3.61730608117724455196267308750, −2.54192046461734572236181467938, −1.56506872964314170475827452754, −0.844753228739777111056501543772,
0.844753228739777111056501543772, 1.56506872964314170475827452754, 2.54192046461734572236181467938, 3.61730608117724455196267308750, 4.45755962899748770496251019839, 5.56427106329783167101003842954, 6.04753499543911157371423859918, 6.49129224933206391529532127711, 7.42529020217861739369672005166, 8.055208100493187914187647428019
