| L(s) = 1 | + 3.11·3-s − 3.70·5-s + 4.20·7-s + 6.70·9-s − 1.09·11-s + 13-s − 11.5·15-s + 0.298·17-s + 1.09·19-s + 13.1·21-s + 8.70·25-s + 11.5·27-s − 2·29-s − 5.13·31-s − 3.40·33-s − 15.5·35-s − 3.70·37-s + 3.11·39-s − 9.40·41-s − 5.29·43-s − 24.8·45-s + 4.20·47-s + 10.7·49-s + 0.929·51-s + 1.40·53-s + 4.04·55-s + 3.40·57-s + ⋯ |
| L(s) = 1 | + 1.79·3-s − 1.65·5-s + 1.59·7-s + 2.23·9-s − 0.329·11-s + 0.277·13-s − 2.97·15-s + 0.0723·17-s + 0.250·19-s + 2.85·21-s + 1.74·25-s + 2.21·27-s − 0.371·29-s − 0.922·31-s − 0.592·33-s − 2.63·35-s − 0.608·37-s + 0.498·39-s − 1.46·41-s − 0.808·43-s − 3.69·45-s + 0.613·47-s + 1.52·49-s + 0.130·51-s + 0.192·53-s + 0.545·55-s + 0.450·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.192573787\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.192573787\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 - 3.11T + 3T^{2} \) |
| 5 | \( 1 + 3.70T + 5T^{2} \) |
| 7 | \( 1 - 4.20T + 7T^{2} \) |
| 11 | \( 1 + 1.09T + 11T^{2} \) |
| 17 | \( 1 - 0.298T + 17T^{2} \) |
| 19 | \( 1 - 1.09T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 5.13T + 31T^{2} \) |
| 37 | \( 1 + 3.70T + 37T^{2} \) |
| 41 | \( 1 + 9.40T + 41T^{2} \) |
| 43 | \( 1 + 5.29T + 43T^{2} \) |
| 47 | \( 1 - 4.20T + 47T^{2} \) |
| 53 | \( 1 - 1.40T + 53T^{2} \) |
| 59 | \( 1 + 13.5T + 59T^{2} \) |
| 61 | \( 1 - 9.40T + 61T^{2} \) |
| 67 | \( 1 + 11.3T + 67T^{2} \) |
| 71 | \( 1 - 8.25T + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 + 14.6T + 79T^{2} \) |
| 83 | \( 1 - 7.32T + 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 - 8.80T + 97T^{2} \) |
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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
−11.27144267912799668933374417498, −10.31740728968048317657341225980, −8.894848668387295020147989288852, −8.408628391007486813206208698121, −7.70983541854264640308889868248, −7.23772232511848076950745793511, −4.95922426227104246141752968546, −4.04611840089777690187100592119, −3.20793014823390904891839771651, −1.72716761869207738952843222092,
1.72716761869207738952843222092, 3.20793014823390904891839771651, 4.04611840089777690187100592119, 4.95922426227104246141752968546, 7.23772232511848076950745793511, 7.70983541854264640308889868248, 8.408628391007486813206208698121, 8.894848668387295020147989288852, 10.31740728968048317657341225980, 11.27144267912799668933374417498
