| L(s) = 1 | − 2-s + 4-s + 2·5-s − 7-s − 8-s − 2·10-s + 4·11-s + 6·13-s + 14-s + 16-s − 2·17-s − 4·19-s + 2·20-s − 4·22-s − 8·23-s − 25-s − 6·26-s − 28-s + 2·29-s − 32-s + 2·34-s − 2·35-s − 10·37-s + 4·38-s − 2·40-s + 6·41-s − 4·43-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.894·5-s − 0.377·7-s − 0.353·8-s − 0.632·10-s + 1.20·11-s + 1.66·13-s + 0.267·14-s + 1/4·16-s − 0.485·17-s − 0.917·19-s + 0.447·20-s − 0.852·22-s − 1.66·23-s − 1/5·25-s − 1.17·26-s − 0.188·28-s + 0.371·29-s − 0.176·32-s + 0.342·34-s − 0.338·35-s − 1.64·37-s + 0.648·38-s − 0.316·40-s + 0.937·41-s − 0.609·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9104737368\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9104737368\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 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 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−13.48577504303344657036010789975, −12.25720048996309464371398882337, −11.12015185557948517225700432181, −10.13805865622749455803240285872, −9.164662289745378790951914691450, −8.337652051623380123011624806396, −6.58813712157544804919343144385, −6.01899647743800981343957937964, −3.83863171110134241641513411613, −1.79365373540820646322667624458,
1.79365373540820646322667624458, 3.83863171110134241641513411613, 6.01899647743800981343957937964, 6.58813712157544804919343144385, 8.337652051623380123011624806396, 9.164662289745378790951914691450, 10.13805865622749455803240285872, 11.12015185557948517225700432181, 12.25720048996309464371398882337, 13.48577504303344657036010789975
