| L(s) = 1 | + 5-s + 2.70·7-s − 0.701·11-s − 13-s + 0.701·17-s − 2·19-s − 6.70·23-s + 25-s − 9.40·29-s − 9.40·31-s + 2.70·35-s − 6.70·37-s − 10.7·41-s + 4·43-s + 1.40·47-s + 0.298·49-s − 10.7·53-s − 0.701·55-s + 14.8·59-s + 2.70·61-s − 65-s − 4·67-s − 15.5·71-s + 13.4·73-s − 1.89·77-s + 4.70·79-s − 4·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.02·7-s − 0.211·11-s − 0.277·13-s + 0.170·17-s − 0.458·19-s − 1.39·23-s + 0.200·25-s − 1.74·29-s − 1.68·31-s + 0.456·35-s − 1.10·37-s − 1.67·41-s + 0.609·43-s + 0.204·47-s + 0.0426·49-s − 1.46·53-s − 0.0945·55-s + 1.92·59-s + 0.345·61-s − 0.124·65-s − 0.488·67-s − 1.84·71-s + 1.56·73-s − 0.215·77-s + 0.528·79-s − 0.439·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 - 2.70T + 7T^{2} \) |
| 11 | \( 1 + 0.701T + 11T^{2} \) |
| 17 | \( 1 - 0.701T + 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 + 6.70T + 23T^{2} \) |
| 29 | \( 1 + 9.40T + 29T^{2} \) |
| 31 | \( 1 + 9.40T + 31T^{2} \) |
| 37 | \( 1 + 6.70T + 37T^{2} \) |
| 41 | \( 1 + 10.7T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 - 1.40T + 47T^{2} \) |
| 53 | \( 1 + 10.7T + 53T^{2} \) |
| 59 | \( 1 - 14.8T + 59T^{2} \) |
| 61 | \( 1 - 2.70T + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 + 15.5T + 71T^{2} \) |
| 73 | \( 1 - 13.4T + 73T^{2} \) |
| 79 | \( 1 - 4.70T + 79T^{2} \) |
| 83 | \( 1 + 4T + 83T^{2} \) |
| 89 | \( 1 + 8.10T + 89T^{2} \) |
| 97 | \( 1 - 18.1T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.892006613504528450388810357080, −7.35917306237466150841656035685, −6.48300439122387768498799826292, −5.53879044653132048238504804967, −5.19038544029302376278141922231, −4.17752139977331914424681958086, −3.42471095993719143067131517811, −2.07745834011207226671191580210, −1.71169458642001441712648222698, 0,
1.71169458642001441712648222698, 2.07745834011207226671191580210, 3.42471095993719143067131517811, 4.17752139977331914424681958086, 5.19038544029302376278141922231, 5.53879044653132048238504804967, 6.48300439122387768498799826292, 7.35917306237466150841656035685, 7.892006613504528450388810357080
