| L(s) = 1 | − 3.23·5-s + 7-s − 2·11-s + 4.47·13-s + 3.23·17-s − 19-s + 2·23-s + 5.47·25-s − 7.70·29-s − 10.4·31-s − 3.23·35-s − 0.472·37-s + 0.472·41-s + 8·43-s − 11.7·47-s + 49-s + 6.76·53-s + 6.47·55-s + 8.94·59-s + 13.4·61-s − 14.4·65-s − 10.4·67-s − 5.70·71-s + 3.52·73-s − 2·77-s + 4·79-s − 0.291·83-s + ⋯ |
| L(s) = 1 | − 1.44·5-s + 0.377·7-s − 0.603·11-s + 1.24·13-s + 0.784·17-s − 0.229·19-s + 0.417·23-s + 1.09·25-s − 1.43·29-s − 1.88·31-s − 0.546·35-s − 0.0776·37-s + 0.0737·41-s + 1.21·43-s − 1.70·47-s + 0.142·49-s + 0.929·53-s + 0.872·55-s + 1.16·59-s + 1.71·61-s − 1.79·65-s − 1.27·67-s − 0.677·71-s + 0.412·73-s − 0.227·77-s + 0.450·79-s − 0.0320·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 + 3.23T + 5T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 - 4.47T + 13T^{2} \) |
| 17 | \( 1 - 3.23T + 17T^{2} \) |
| 23 | \( 1 - 2T + 23T^{2} \) |
| 29 | \( 1 + 7.70T + 29T^{2} \) |
| 31 | \( 1 + 10.4T + 31T^{2} \) |
| 37 | \( 1 + 0.472T + 37T^{2} \) |
| 41 | \( 1 - 0.472T + 41T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 + 11.7T + 47T^{2} \) |
| 53 | \( 1 - 6.76T + 53T^{2} \) |
| 59 | \( 1 - 8.94T + 59T^{2} \) |
| 61 | \( 1 - 13.4T + 61T^{2} \) |
| 67 | \( 1 + 10.4T + 67T^{2} \) |
| 71 | \( 1 + 5.70T + 71T^{2} \) |
| 73 | \( 1 - 3.52T + 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 + 0.291T + 83T^{2} \) |
| 89 | \( 1 + 4.47T + 89T^{2} \) |
| 97 | \( 1 - 9.41T + 97T^{2} \) |
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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
−7.44501949586773865423740920188, −6.92393248167170013205876388661, −5.80659710904386742594266225058, −5.36617909703670805193324771814, −4.42663782277204671861934277480, −3.69024014555292146124898775045, −3.38149204607146716231062852008, −2.15370415907958461133147686694, −1.09496649740619004624668896937, 0,
1.09496649740619004624668896937, 2.15370415907958461133147686694, 3.38149204607146716231062852008, 3.69024014555292146124898775045, 4.42663782277204671861934277480, 5.36617909703670805193324771814, 5.80659710904386742594266225058, 6.92393248167170013205876388661, 7.44501949586773865423740920188
