| L(s) = 1 | − 5-s − 3.90·7-s − 6.09·11-s − 2.02·13-s + 2.46·17-s − 3.33·19-s + 0.830·23-s + 25-s − 2.63·29-s − 1.83·31-s + 3.90·35-s + 5.58·37-s − 6.65·41-s − 11.0·43-s + 0.166·47-s + 8.21·49-s − 12.0·53-s + 6.09·55-s − 1.73·59-s − 13.3·61-s + 2.02·65-s − 13.2·67-s + 10.8·71-s + 1.87·73-s + 23.7·77-s − 7.46·79-s + 6.80·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.47·7-s − 1.83·11-s − 0.561·13-s + 0.597·17-s − 0.765·19-s + 0.173·23-s + 0.200·25-s − 0.488·29-s − 0.328·31-s + 0.659·35-s + 0.918·37-s − 1.03·41-s − 1.68·43-s + 0.0243·47-s + 1.17·49-s − 1.65·53-s + 0.821·55-s − 0.225·59-s − 1.70·61-s + 0.251·65-s − 1.62·67-s + 1.29·71-s + 0.219·73-s + 2.70·77-s − 0.839·79-s + 0.746·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3425050193\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3425050193\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 3.90T + 7T^{2} \) |
| 11 | \( 1 + 6.09T + 11T^{2} \) |
| 13 | \( 1 + 2.02T + 13T^{2} \) |
| 17 | \( 1 - 2.46T + 17T^{2} \) |
| 19 | \( 1 + 3.33T + 19T^{2} \) |
| 23 | \( 1 - 0.830T + 23T^{2} \) |
| 29 | \( 1 + 2.63T + 29T^{2} \) |
| 31 | \( 1 + 1.83T + 31T^{2} \) |
| 37 | \( 1 - 5.58T + 37T^{2} \) |
| 41 | \( 1 + 6.65T + 41T^{2} \) |
| 43 | \( 1 + 11.0T + 43T^{2} \) |
| 47 | \( 1 - 0.166T + 47T^{2} \) |
| 53 | \( 1 + 12.0T + 53T^{2} \) |
| 59 | \( 1 + 1.73T + 59T^{2} \) |
| 61 | \( 1 + 13.3T + 61T^{2} \) |
| 67 | \( 1 + 13.2T + 67T^{2} \) |
| 71 | \( 1 - 10.8T + 71T^{2} \) |
| 73 | \( 1 - 1.87T + 73T^{2} \) |
| 79 | \( 1 + 7.46T + 79T^{2} \) |
| 83 | \( 1 - 6.80T + 83T^{2} \) |
| 89 | \( 1 + 12.1T + 89T^{2} \) |
| 97 | \( 1 + 0.663T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.86336937741875505961996315996, −7.43508344110739450090352116855, −6.59560727444224295537777421712, −5.95538229331737460575190339364, −5.14117198837266932656179427163, −4.47701478353569070213202650668, −3.26072178120698813808410193441, −3.05674857242172317035079996890, −1.97389227441624471752744031581, −0.28019659636436873962703866560,
0.28019659636436873962703866560, 1.97389227441624471752744031581, 3.05674857242172317035079996890, 3.26072178120698813808410193441, 4.47701478353569070213202650668, 5.14117198837266932656179427163, 5.95538229331737460575190339364, 6.59560727444224295537777421712, 7.43508344110739450090352116855, 7.86336937741875505961996315996
