| L(s) = 1 | − 4·5-s + 3·13-s + 4·17-s + 7·19-s − 4·23-s + 11·25-s − 8·29-s − 5·31-s + 3·37-s − 8·41-s − 11·43-s + 4·47-s + 4·53-s + 12·59-s + 2·61-s − 12·65-s + 3·67-s − 12·71-s − 73-s − 79-s + 12·83-s − 16·85-s − 8·89-s − 28·95-s + 2·97-s + 3·103-s + 12·107-s + ⋯ |
| L(s) = 1 | − 1.78·5-s + 0.832·13-s + 0.970·17-s + 1.60·19-s − 0.834·23-s + 11/5·25-s − 1.48·29-s − 0.898·31-s + 0.493·37-s − 1.24·41-s − 1.67·43-s + 0.583·47-s + 0.549·53-s + 1.56·59-s + 0.256·61-s − 1.48·65-s + 0.366·67-s − 1.42·71-s − 0.117·73-s − 0.112·79-s + 1.31·83-s − 1.73·85-s − 0.847·89-s − 2.87·95-s + 0.203·97-s + 0.295·103-s + 1.16·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.198241711\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.198241711\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−7.902411121628635971953033958212, −7.37786121519322206420381337939, −6.79574495224184795847290893333, −5.66563702886740437379671841736, −5.17520107670724123245452546978, −4.08173413639678390137451260087, −3.61646063426785494401781033005, −3.10595913766941266781565341125, −1.62894710985601508842847294311, −0.57426866169805545305079539621,
0.57426866169805545305079539621, 1.62894710985601508842847294311, 3.10595913766941266781565341125, 3.61646063426785494401781033005, 4.08173413639678390137451260087, 5.17520107670724123245452546978, 5.66563702886740437379671841736, 6.79574495224184795847290893333, 7.37786121519322206420381337939, 7.902411121628635971953033958212
