| L(s) = 1 | − 3-s + 5-s + 9-s + 11-s − 13-s − 15-s + 6·17-s + 7·19-s − 4·23-s − 4·25-s − 27-s + 3·29-s + 2·31-s − 33-s − 7·37-s + 39-s − 8·41-s + 45-s − 13·47-s − 6·51-s − 12·53-s + 55-s − 7·57-s − 59-s − 14·61-s − 65-s − 15·67-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1/3·9-s + 0.301·11-s − 0.277·13-s − 0.258·15-s + 1.45·17-s + 1.60·19-s − 0.834·23-s − 4/5·25-s − 0.192·27-s + 0.557·29-s + 0.359·31-s − 0.174·33-s − 1.15·37-s + 0.160·39-s − 1.24·41-s + 0.149·45-s − 1.89·47-s − 0.840·51-s − 1.64·53-s + 0.134·55-s − 0.927·57-s − 0.130·59-s − 1.79·61-s − 0.124·65-s − 1.83·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 5 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 13 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 15 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
| 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
−16.59437387473646, −15.95550744645812, −15.68165262102623, −14.71988849830779, −14.30611485625301, −13.67278179853282, −13.32529091664321, −12.29602735993889, −11.96857899241970, −11.69845158133888, −10.73709808613316, −10.10252847098267, −9.791689536797268, −9.222544197068137, −8.270126564455656, −7.681525538253869, −7.175348441435996, −6.194583558080703, −5.956929552326690, −5.059450773582710, −4.712975719721971, −3.468605279501098, −3.160109583287761, −1.813476172268078, −1.270217910891677, 0,
1.270217910891677, 1.813476172268078, 3.160109583287761, 3.468605279501098, 4.712975719721971, 5.059450773582710, 5.956929552326690, 6.194583558080703, 7.175348441435996, 7.681525538253869, 8.270126564455656, 9.222544197068137, 9.791689536797268, 10.10252847098267, 10.73709808613316, 11.69845158133888, 11.96857899241970, 12.29602735993889, 13.32529091664321, 13.67278179853282, 14.30611485625301, 14.71988849830779, 15.68165262102623, 15.95550744645812, 16.59437387473646
