| L(s) = 1 | + 2.94·5-s + 4.65·7-s + 5.56·11-s − 0.393·13-s + 0.257·17-s + 2.11·19-s − 5.38·23-s + 3.68·25-s + 0.101·29-s − 3.82·31-s + 13.7·35-s − 1.36·37-s + 5.79·41-s + 7.94·43-s − 5.64·47-s + 14.6·49-s + 4.37·53-s + 16.3·55-s − 7.40·59-s + 4.23·61-s − 1.15·65-s − 4.79·67-s + 7.11·71-s + 9.79·73-s + 25.8·77-s − 4.39·79-s − 9.64·83-s + ⋯ |
| L(s) = 1 | + 1.31·5-s + 1.75·7-s + 1.67·11-s − 0.109·13-s + 0.0625·17-s + 0.484·19-s − 1.12·23-s + 0.737·25-s + 0.0187·29-s − 0.686·31-s + 2.31·35-s − 0.224·37-s + 0.904·41-s + 1.21·43-s − 0.822·47-s + 2.09·49-s + 0.600·53-s + 2.21·55-s − 0.963·59-s + 0.541·61-s − 0.143·65-s − 0.586·67-s + 0.844·71-s + 1.14·73-s + 2.94·77-s − 0.494·79-s − 1.05·83-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
| \(L(1)\) |
\(\approx\) |
\(3.858801307\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.858801307\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.94T + 5T^{2} \) |
| 7 | \( 1 - 4.65T + 7T^{2} \) |
| 11 | \( 1 - 5.56T + 11T^{2} \) |
| 13 | \( 1 + 0.393T + 13T^{2} \) |
| 17 | \( 1 - 0.257T + 17T^{2} \) |
| 19 | \( 1 - 2.11T + 19T^{2} \) |
| 23 | \( 1 + 5.38T + 23T^{2} \) |
| 29 | \( 1 - 0.101T + 29T^{2} \) |
| 31 | \( 1 + 3.82T + 31T^{2} \) |
| 37 | \( 1 + 1.36T + 37T^{2} \) |
| 41 | \( 1 - 5.79T + 41T^{2} \) |
| 43 | \( 1 - 7.94T + 43T^{2} \) |
| 47 | \( 1 + 5.64T + 47T^{2} \) |
| 53 | \( 1 - 4.37T + 53T^{2} \) |
| 59 | \( 1 + 7.40T + 59T^{2} \) |
| 61 | \( 1 - 4.23T + 61T^{2} \) |
| 67 | \( 1 + 4.79T + 67T^{2} \) |
| 71 | \( 1 - 7.11T + 71T^{2} \) |
| 73 | \( 1 - 9.79T + 73T^{2} \) |
| 79 | \( 1 + 4.39T + 79T^{2} \) |
| 83 | \( 1 + 9.64T + 83T^{2} \) |
| 89 | \( 1 + 10.6T + 89T^{2} \) |
| 97 | \( 1 + 5.54T + 97T^{2} \) |
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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
−8.117348618703694299582950499948, −7.39130278507874623380708517611, −6.59280829936479685237279438737, −5.82186765592708723253930181873, −5.36040016999124551855602403079, −4.44329060265106933894488196753, −3.84778560100381619706305540260, −2.48215525152398914552305103720, −1.70627983744025703731735245679, −1.21156871219022255999254237124,
1.21156871219022255999254237124, 1.70627983744025703731735245679, 2.48215525152398914552305103720, 3.84778560100381619706305540260, 4.44329060265106933894488196753, 5.36040016999124551855602403079, 5.82186765592708723253930181873, 6.59280829936479685237279438737, 7.39130278507874623380708517611, 8.117348618703694299582950499948
