| L(s) = 1 | + 2.67·3-s + 5-s + 1.14·7-s + 4.14·9-s − 3.81·11-s + 13-s + 2.67·15-s + 7.34·17-s − 5.52·19-s + 3.05·21-s + 6.67·23-s + 25-s + 3.05·27-s + 2.85·29-s + 0.183·31-s − 10.2·33-s + 1.14·35-s + 6.48·37-s + 2.67·39-s − 7.34·41-s − 2.67·43-s + 4.14·45-s + 2.85·47-s − 5.69·49-s + 19.6·51-s − 13.6·53-s − 3.81·55-s + ⋯ |
| L(s) = 1 | + 1.54·3-s + 0.447·5-s + 0.432·7-s + 1.38·9-s − 1.15·11-s + 0.277·13-s + 0.690·15-s + 1.78·17-s − 1.26·19-s + 0.667·21-s + 1.39·23-s + 0.200·25-s + 0.588·27-s + 0.530·29-s + 0.0329·31-s − 1.77·33-s + 0.193·35-s + 1.06·37-s + 0.427·39-s − 1.14·41-s − 0.407·43-s + 0.617·45-s + 0.416·47-s − 0.813·49-s + 2.74·51-s − 1.87·53-s − 0.514·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.013539680\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.013539680\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 - 2.67T + 3T^{2} \) |
| 7 | \( 1 - 1.14T + 7T^{2} \) |
| 11 | \( 1 + 3.81T + 11T^{2} \) |
| 17 | \( 1 - 7.34T + 17T^{2} \) |
| 19 | \( 1 + 5.52T + 19T^{2} \) |
| 23 | \( 1 - 6.67T + 23T^{2} \) |
| 29 | \( 1 - 2.85T + 29T^{2} \) |
| 31 | \( 1 - 0.183T + 31T^{2} \) |
| 37 | \( 1 - 6.48T + 37T^{2} \) |
| 41 | \( 1 + 7.34T + 41T^{2} \) |
| 43 | \( 1 + 2.67T + 43T^{2} \) |
| 47 | \( 1 - 2.85T + 47T^{2} \) |
| 53 | \( 1 + 13.6T + 53T^{2} \) |
| 59 | \( 1 + 5.16T + 59T^{2} \) |
| 61 | \( 1 - 5.14T + 61T^{2} \) |
| 67 | \( 1 - 2.48T + 67T^{2} \) |
| 71 | \( 1 + 3.81T + 71T^{2} \) |
| 73 | \( 1 - 12.7T + 73T^{2} \) |
| 79 | \( 1 + 9.34T + 79T^{2} \) |
| 83 | \( 1 + 11.8T + 83T^{2} \) |
| 89 | \( 1 + 14.9T + 89T^{2} \) |
| 97 | \( 1 - 7.71T + 97T^{2} \) |
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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
−9.850336880705064798793101830614, −8.998559993636642324401350974779, −8.162562645788088439918627877661, −7.84560122179873871791640574089, −6.73662887032070638582883621928, −5.52583891608374808763382940728, −4.60097703511887381589523409145, −3.32071845569466894375693843157, −2.66260455796779731719241242320, −1.50990287266067694004263976832,
1.50990287266067694004263976832, 2.66260455796779731719241242320, 3.32071845569466894375693843157, 4.60097703511887381589523409145, 5.52583891608374808763382940728, 6.73662887032070638582883621928, 7.84560122179873871791640574089, 8.162562645788088439918627877661, 8.998559993636642324401350974779, 9.850336880705064798793101830614
