| L(s) = 1 | − 2.29·5-s + 0.183·7-s + 5.49·11-s + 0.551·13-s − 4.19·17-s + 19-s − 1.55·23-s + 0.258·25-s − 6.12·29-s + 0.381·31-s − 0.420·35-s + 9.05·37-s − 2.94·41-s − 2.53·43-s − 11.3·47-s − 6.96·49-s + 11.2·53-s − 12.6·55-s + 5.13·59-s + 6.93·61-s − 1.26·65-s − 7.74·67-s + 13.0·71-s − 9.71·73-s + 1.00·77-s + 10.0·79-s − 11.0·83-s + ⋯ |
| L(s) = 1 | − 1.02·5-s + 0.0692·7-s + 1.65·11-s + 0.153·13-s − 1.01·17-s + 0.229·19-s − 0.324·23-s + 0.0517·25-s − 1.13·29-s + 0.0685·31-s − 0.0710·35-s + 1.48·37-s − 0.460·41-s − 0.386·43-s − 1.65·47-s − 0.995·49-s + 1.54·53-s − 1.69·55-s + 0.668·59-s + 0.887·61-s − 0.156·65-s − 0.945·67-s + 1.54·71-s − 1.13·73-s + 0.114·77-s + 1.12·79-s − 1.20·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8208 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 + 2.29T + 5T^{2} \) |
| 7 | \( 1 - 0.183T + 7T^{2} \) |
| 11 | \( 1 - 5.49T + 11T^{2} \) |
| 13 | \( 1 - 0.551T + 13T^{2} \) |
| 17 | \( 1 + 4.19T + 17T^{2} \) |
| 23 | \( 1 + 1.55T + 23T^{2} \) |
| 29 | \( 1 + 6.12T + 29T^{2} \) |
| 31 | \( 1 - 0.381T + 31T^{2} \) |
| 37 | \( 1 - 9.05T + 37T^{2} \) |
| 41 | \( 1 + 2.94T + 41T^{2} \) |
| 43 | \( 1 + 2.53T + 43T^{2} \) |
| 47 | \( 1 + 11.3T + 47T^{2} \) |
| 53 | \( 1 - 11.2T + 53T^{2} \) |
| 59 | \( 1 - 5.13T + 59T^{2} \) |
| 61 | \( 1 - 6.93T + 61T^{2} \) |
| 67 | \( 1 + 7.74T + 67T^{2} \) |
| 71 | \( 1 - 13.0T + 71T^{2} \) |
| 73 | \( 1 + 9.71T + 73T^{2} \) |
| 79 | \( 1 - 10.0T + 79T^{2} \) |
| 83 | \( 1 + 11.0T + 83T^{2} \) |
| 89 | \( 1 - 3.76T + 89T^{2} \) |
| 97 | \( 1 - 4.23T + 97T^{2} \) |
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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
−7.45768530195886996108233046126, −6.76236769033173666881260099029, −6.29302276089872811936493260744, −5.34805885242533692614807598943, −4.37913923908325123617572249846, −3.97218701727335815223412442169, −3.33090728163136241189051242226, −2.15577940420614214586889513052, −1.20814932686571511626329824953, 0,
1.20814932686571511626329824953, 2.15577940420614214586889513052, 3.33090728163136241189051242226, 3.97218701727335815223412442169, 4.37913923908325123617572249846, 5.34805885242533692614807598943, 6.29302276089872811936493260744, 6.76236769033173666881260099029, 7.45768530195886996108233046126
