| L(s) = 1 | + 3.32·5-s − 2.77·7-s − 2.53·11-s + 0.745·13-s − 4.83·17-s + 19-s + 1.29·23-s + 6.07·25-s + 2.01·29-s − 1.93·31-s − 9.23·35-s − 1.83·37-s + 11.7·41-s − 5.64·43-s − 4.39·47-s + 0.707·49-s + 1.00·53-s − 8.42·55-s + 0.602·59-s + 4.18·61-s + 2.48·65-s − 1.92·67-s + 0.548·71-s − 5.91·73-s + 7.03·77-s − 8.96·79-s + 11.0·83-s + ⋯ |
| L(s) = 1 | + 1.48·5-s − 1.04·7-s − 0.763·11-s + 0.206·13-s − 1.17·17-s + 0.229·19-s + 0.270·23-s + 1.21·25-s + 0.373·29-s − 0.348·31-s − 1.56·35-s − 0.300·37-s + 1.84·41-s − 0.860·43-s − 0.641·47-s + 0.101·49-s + 0.138·53-s − 1.13·55-s + 0.0784·59-s + 0.535·61-s + 0.307·65-s − 0.235·67-s + 0.0651·71-s − 0.692·73-s + 0.801·77-s − 1.00·79-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 | \( 1 - 3.32T + 5T^{2} \) |
| 7 | \( 1 + 2.77T + 7T^{2} \) |
| 11 | \( 1 + 2.53T + 11T^{2} \) |
| 13 | \( 1 - 0.745T + 13T^{2} \) |
| 17 | \( 1 + 4.83T + 17T^{2} \) |
| 23 | \( 1 - 1.29T + 23T^{2} \) |
| 29 | \( 1 - 2.01T + 29T^{2} \) |
| 31 | \( 1 + 1.93T + 31T^{2} \) |
| 37 | \( 1 + 1.83T + 37T^{2} \) |
| 41 | \( 1 - 11.7T + 41T^{2} \) |
| 43 | \( 1 + 5.64T + 43T^{2} \) |
| 47 | \( 1 + 4.39T + 47T^{2} \) |
| 53 | \( 1 - 1.00T + 53T^{2} \) |
| 59 | \( 1 - 0.602T + 59T^{2} \) |
| 61 | \( 1 - 4.18T + 61T^{2} \) |
| 67 | \( 1 + 1.92T + 67T^{2} \) |
| 71 | \( 1 - 0.548T + 71T^{2} \) |
| 73 | \( 1 + 5.91T + 73T^{2} \) |
| 79 | \( 1 + 8.96T + 79T^{2} \) |
| 83 | \( 1 - 11.0T + 83T^{2} \) |
| 89 | \( 1 + 10.4T + 89T^{2} \) |
| 97 | \( 1 - 16.0T + 97T^{2} \) |
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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
−7.29768967612337460834491484175, −6.59963835976103464701565363364, −6.16167666280810530412538403514, −5.48119680239079329385491755616, −4.83082831689696252008195310376, −3.83144739891477382224664985174, −2.82858499885689268201405031371, −2.38154724079017977901233900583, −1.36273996462024888047406897890, 0,
1.36273996462024888047406897890, 2.38154724079017977901233900583, 2.82858499885689268201405031371, 3.83144739891477382224664985174, 4.83082831689696252008195310376, 5.48119680239079329385491755616, 6.16167666280810530412538403514, 6.59963835976103464701565363364, 7.29768967612337460834491484175
