| L(s) = 1 | + 2-s − 3-s + 4-s − 2·5-s − 6-s − 4·7-s + 8-s + 9-s − 2·10-s − 11-s − 12-s − 4·14-s + 2·15-s + 16-s − 17-s + 18-s − 4·19-s − 2·20-s + 4·21-s − 22-s − 24-s − 25-s − 27-s − 4·28-s + 2·29-s + 2·30-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 − 1.51·7-s + 0.353·8-s + 1/3·9-s − 0.632·10-s − 0.301·11-s − 0.288·12-s − 1.06·14-s + 0.516·15-s + 1/4·16-s − 0.242·17-s + 0.235·18-s − 0.917·19-s − 0.447·20-s + 0.872·21-s − 0.213·22-s − 0.204·24-s − 1/5·25-s − 0.192·27-s − 0.755·28-s + 0.371·29-s + 0.365·30-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189618 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189618 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.118437415\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.118437415\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−13.01405252693218, −12.65693602873522, −12.30160726573475, −11.76680861398480, −11.31046708991283, −10.90382087544122, −10.38041514335458, −9.930766184958047, −9.394408640320890, −8.923353272346501, −8.147962409493779, −7.729080002438609, −7.279085741841911, −6.622322030639193, −6.222149583794654, −6.042542487843016, −5.243023078618889, −4.633315884012958, −4.220161569270971, −3.706725739969240, −3.219854354717375, −2.597578542201910, −2.058432903637685, −0.9659904988375512, −0.3229225669928622,
0.3229225669928622, 0.9659904988375512, 2.058432903637685, 2.597578542201910, 3.219854354717375, 3.706725739969240, 4.220161569270971, 4.633315884012958, 5.243023078618889, 6.042542487843016, 6.222149583794654, 6.622322030639193, 7.279085741841911, 7.729080002438609, 8.147962409493779, 8.923353272346501, 9.394408640320890, 9.930766184958047, 10.38041514335458, 10.90382087544122, 11.31046708991283, 11.76680861398480, 12.30160726573475, 12.65693602873522, 13.01405252693218
