| L(s) = 1 | − 2·5-s − 0.516·7-s − 0.538·11-s − 0.295·13-s + 6.99·17-s − 4.06·19-s − 0.781·23-s − 25-s − 0.811·29-s + 3.07·31-s + 1.03·35-s − 3.95·37-s − 0.743·41-s + 0.590·43-s + 3.47·47-s − 6.73·49-s − 6.69·53-s + 1.07·55-s + 8.91·59-s + 4.63·61-s + 0.590·65-s + 6.55·67-s − 12.4·71-s + 7.77·73-s + 0.278·77-s − 4.88·79-s + 1.11·83-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 0.195·7-s − 0.162·11-s − 0.0819·13-s + 1.69·17-s − 0.932·19-s − 0.162·23-s − 0.200·25-s − 0.150·29-s + 0.552·31-s + 0.174·35-s − 0.650·37-s − 0.116·41-s + 0.0900·43-s + 0.506·47-s − 0.961·49-s − 0.920·53-s + 0.145·55-s + 1.16·59-s + 0.594·61-s + 0.0732·65-s + 0.800·67-s − 1.47·71-s + 0.910·73-s + 0.0316·77-s − 0.549·79-s + 0.122·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.267664167\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.267664167\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 167 | \( 1 - T \) |
| good | 5 | \( 1 + 2T + 5T^{2} \) |
| 7 | \( 1 + 0.516T + 7T^{2} \) |
| 11 | \( 1 + 0.538T + 11T^{2} \) |
| 13 | \( 1 + 0.295T + 13T^{2} \) |
| 17 | \( 1 - 6.99T + 17T^{2} \) |
| 19 | \( 1 + 4.06T + 19T^{2} \) |
| 23 | \( 1 + 0.781T + 23T^{2} \) |
| 29 | \( 1 + 0.811T + 29T^{2} \) |
| 31 | \( 1 - 3.07T + 31T^{2} \) |
| 37 | \( 1 + 3.95T + 37T^{2} \) |
| 41 | \( 1 + 0.743T + 41T^{2} \) |
| 43 | \( 1 - 0.590T + 43T^{2} \) |
| 47 | \( 1 - 3.47T + 47T^{2} \) |
| 53 | \( 1 + 6.69T + 53T^{2} \) |
| 59 | \( 1 - 8.91T + 59T^{2} \) |
| 61 | \( 1 - 4.63T + 61T^{2} \) |
| 67 | \( 1 - 6.55T + 67T^{2} \) |
| 71 | \( 1 + 12.4T + 71T^{2} \) |
| 73 | \( 1 - 7.77T + 73T^{2} \) |
| 79 | \( 1 + 4.88T + 79T^{2} \) |
| 83 | \( 1 - 1.11T + 83T^{2} \) |
| 89 | \( 1 - 1.79T + 89T^{2} \) |
| 97 | \( 1 + 17.1T + 97T^{2} \) |
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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
−8.070537639683897164836786744980, −7.47946161093326635449360600193, −6.74748013498605413466388370515, −5.93014195926811040038089278188, −5.21371941611930490532709229290, −4.33653068101832150671045387553, −3.64165772659627140897306247006, −2.96358500094648538473670534937, −1.82388446491274241274680364875, −0.58624321435360987084695345712,
0.58624321435360987084695345712, 1.82388446491274241274680364875, 2.96358500094648538473670534937, 3.64165772659627140897306247006, 4.33653068101832150671045387553, 5.21371941611930490532709229290, 5.93014195926811040038089278188, 6.74748013498605413466388370515, 7.47946161093326635449360600193, 8.070537639683897164836786744980
