| L(s) = 1 | + 2.60·3-s + 5-s + 2.76·7-s + 3.76·9-s + 2.16·11-s + 2.60·15-s + 5.03·19-s + 7.20·21-s + 4.93·23-s + 25-s + 2.00·27-s − 4.43·29-s − 3.37·31-s + 5.63·33-s + 2.76·35-s − 3.97·37-s − 1.66·41-s − 9.80·43-s + 3.76·45-s + 11.6·47-s + 0.665·49-s − 12.7·53-s + 2.16·55-s + 13.1·57-s − 8.16·59-s − 10.3·61-s + 10.4·63-s + ⋯ |
| L(s) = 1 | + 1.50·3-s + 0.447·5-s + 1.04·7-s + 1.25·9-s + 0.653·11-s + 0.671·15-s + 1.15·19-s + 1.57·21-s + 1.02·23-s + 0.200·25-s + 0.384·27-s − 0.823·29-s − 0.605·31-s + 0.981·33-s + 0.468·35-s − 0.653·37-s − 0.260·41-s − 1.49·43-s + 0.561·45-s + 1.69·47-s + 0.0951·49-s − 1.75·53-s + 0.292·55-s + 1.73·57-s − 1.06·59-s − 1.31·61-s + 1.31·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.333875163\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.333875163\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 3 | \( 1 - 2.60T + 3T^{2} \) |
| 7 | \( 1 - 2.76T + 7T^{2} \) |
| 11 | \( 1 - 2.16T + 11T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 5.03T + 19T^{2} \) |
| 23 | \( 1 - 4.93T + 23T^{2} \) |
| 29 | \( 1 + 4.43T + 29T^{2} \) |
| 31 | \( 1 + 3.37T + 31T^{2} \) |
| 37 | \( 1 + 3.97T + 37T^{2} \) |
| 41 | \( 1 + 1.66T + 41T^{2} \) |
| 43 | \( 1 + 9.80T + 43T^{2} \) |
| 47 | \( 1 - 11.6T + 47T^{2} \) |
| 53 | \( 1 + 12.7T + 53T^{2} \) |
| 59 | \( 1 + 8.16T + 59T^{2} \) |
| 61 | \( 1 + 10.3T + 61T^{2} \) |
| 67 | \( 1 + 7.10T + 67T^{2} \) |
| 71 | \( 1 - 16.1T + 71T^{2} \) |
| 73 | \( 1 - 15.9T + 73T^{2} \) |
| 79 | \( 1 - 11.9T + 79T^{2} \) |
| 83 | \( 1 + 3.97T + 83T^{2} \) |
| 89 | \( 1 + 7.94T + 89T^{2} \) |
| 97 | \( 1 - 0.462T + 97T^{2} \) |
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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
−8.673952038725713576442875211455, −7.901225619490832545047571138817, −7.41391524767570049627124251575, −6.56095157822754553106056105817, −5.40823499143385000683210246605, −4.75297502228793133709639180381, −3.68381993221106659668510947671, −3.07341322162719367414542733606, −1.98809386082915146684609802538, −1.36103245414624718849959780079,
1.36103245414624718849959780079, 1.98809386082915146684609802538, 3.07341322162719367414542733606, 3.68381993221106659668510947671, 4.75297502228793133709639180381, 5.40823499143385000683210246605, 6.56095157822754553106056105817, 7.41391524767570049627124251575, 7.901225619490832545047571138817, 8.673952038725713576442875211455
