| L(s) = 1 | + 5-s − 0.561·7-s + 3.56·11-s + 13-s + 6.68·17-s + 7.68·19-s + 3.56·23-s + 25-s − 4.68·29-s + 2.43·31-s − 0.561·35-s − 5.68·37-s − 4·41-s − 4.68·43-s − 3.56·47-s − 6.68·49-s + 3.56·55-s − 1.12·59-s − 9.68·61-s + 65-s + 5.43·67-s − 14.2·71-s + 13.6·73-s − 2·77-s + 17.2·79-s + 4.87·83-s + 6.68·85-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.212·7-s + 1.07·11-s + 0.277·13-s + 1.62·17-s + 1.76·19-s + 0.742·23-s + 0.200·25-s − 0.869·29-s + 0.437·31-s − 0.0949·35-s − 0.934·37-s − 0.624·41-s − 0.714·43-s − 0.519·47-s − 0.954·49-s + 0.480·55-s − 0.146·59-s − 1.23·61-s + 0.124·65-s + 0.664·67-s − 1.69·71-s + 1.60·73-s − 0.227·77-s + 1.94·79-s + 0.535·83-s + 0.725·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.548543285\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.548543285\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| good | 7 | \( 1 + 0.561T + 7T^{2} \) |
| 11 | \( 1 - 3.56T + 11T^{2} \) |
| 13 | \( 1 - T + 13T^{2} \) |
| 17 | \( 1 - 6.68T + 17T^{2} \) |
| 19 | \( 1 - 7.68T + 19T^{2} \) |
| 23 | \( 1 - 3.56T + 23T^{2} \) |
| 29 | \( 1 + 4.68T + 29T^{2} \) |
| 31 | \( 1 - 2.43T + 31T^{2} \) |
| 37 | \( 1 + 5.68T + 37T^{2} \) |
| 41 | \( 1 + 4T + 41T^{2} \) |
| 43 | \( 1 + 4.68T + 43T^{2} \) |
| 47 | \( 1 + 3.56T + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 1.12T + 59T^{2} \) |
| 61 | \( 1 + 9.68T + 61T^{2} \) |
| 67 | \( 1 - 5.43T + 67T^{2} \) |
| 71 | \( 1 + 14.2T + 71T^{2} \) |
| 73 | \( 1 - 13.6T + 73T^{2} \) |
| 79 | \( 1 - 17.2T + 79T^{2} \) |
| 83 | \( 1 - 4.87T + 83T^{2} \) |
| 89 | \( 1 + 11.3T + 89T^{2} \) |
| 97 | \( 1 + 2.31T + 97T^{2} \) |
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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
−8.387802872818166949029886260928, −7.58834916603507781727032148638, −6.93016011757763286073711213239, −6.16192387152083801667633398695, −5.43759713354625233979431478423, −4.79898003030180165640587666914, −3.37308428409870558430536063629, −3.34009387211846778489016871849, −1.74293718312797842196237120454, −0.987597467894842147328965860888,
0.987597467894842147328965860888, 1.74293718312797842196237120454, 3.34009387211846778489016871849, 3.37308428409870558430536063629, 4.79898003030180165640587666914, 5.43759713354625233979431478423, 6.16192387152083801667633398695, 6.93016011757763286073711213239, 7.58834916603507781727032148638, 8.387802872818166949029886260928
