| L(s) = 1 | − 2.58·5-s + 7-s + 4.04·11-s − 0.190·13-s + 6.82·17-s − 1.34·19-s − 0.240·23-s + 1.65·25-s − 0.190·29-s + 2.46·31-s − 2.58·35-s + 10.4·37-s + 2.77·41-s − 11.4·43-s − 4.29·47-s + 49-s − 9.26·53-s − 10.4·55-s + 12.9·59-s − 13.6·61-s + 0.490·65-s − 0.871·67-s + 11.0·71-s + 12.5·73-s + 4.04·77-s − 15.0·79-s + 4.19·83-s + ⋯ |
| L(s) = 1 | − 1.15·5-s + 0.377·7-s + 1.22·11-s − 0.0527·13-s + 1.65·17-s − 0.307·19-s − 0.0500·23-s + 0.331·25-s − 0.0353·29-s + 0.443·31-s − 0.436·35-s + 1.71·37-s + 0.432·41-s − 1.74·43-s − 0.625·47-s + 0.142·49-s − 1.27·53-s − 1.40·55-s + 1.68·59-s − 1.74·61-s + 0.0608·65-s − 0.106·67-s + 1.31·71-s + 1.46·73-s + 0.461·77-s − 1.69·79-s + 0.459·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.806641258\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.806641258\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| good | 5 | \( 1 + 2.58T + 5T^{2} \) |
| 11 | \( 1 - 4.04T + 11T^{2} \) |
| 13 | \( 1 + 0.190T + 13T^{2} \) |
| 17 | \( 1 - 6.82T + 17T^{2} \) |
| 19 | \( 1 + 1.34T + 19T^{2} \) |
| 23 | \( 1 + 0.240T + 23T^{2} \) |
| 29 | \( 1 + 0.190T + 29T^{2} \) |
| 31 | \( 1 - 2.46T + 31T^{2} \) |
| 37 | \( 1 - 10.4T + 37T^{2} \) |
| 41 | \( 1 - 2.77T + 41T^{2} \) |
| 43 | \( 1 + 11.4T + 43T^{2} \) |
| 47 | \( 1 + 4.29T + 47T^{2} \) |
| 53 | \( 1 + 9.26T + 53T^{2} \) |
| 59 | \( 1 - 12.9T + 59T^{2} \) |
| 61 | \( 1 + 13.6T + 61T^{2} \) |
| 67 | \( 1 + 0.871T + 67T^{2} \) |
| 71 | \( 1 - 11.0T + 71T^{2} \) |
| 73 | \( 1 - 12.5T + 73T^{2} \) |
| 79 | \( 1 + 15.0T + 79T^{2} \) |
| 83 | \( 1 - 4.19T + 83T^{2} \) |
| 89 | \( 1 - 10.3T + 89T^{2} \) |
| 97 | \( 1 + 10.3T + 97T^{2} \) |
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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
−8.035923231766130191463837597323, −7.53828348766238353176402733898, −6.69376299240256899032147448857, −6.03687818618251709091688457049, −5.08099565591386451220694979339, −4.32539563792993796321515971365, −3.70060737761899590318465919726, −3.01306509434009332523956426195, −1.67594470274124106960422845604, −0.74367617405895369296108230771,
0.74367617405895369296108230771, 1.67594470274124106960422845604, 3.01306509434009332523956426195, 3.70060737761899590318465919726, 4.32539563792993796321515971365, 5.08099565591386451220694979339, 6.03687818618251709091688457049, 6.69376299240256899032147448857, 7.53828348766238353176402733898, 8.035923231766130191463837597323
