| L(s) = 1 | − 3-s − 2.60·5-s − 1.53·7-s + 9-s + 4.96·11-s + 3.57·13-s + 2.60·15-s − 5.67·17-s + 7.50·19-s + 1.53·21-s − 5.35·23-s + 1.78·25-s − 27-s + 3.66·29-s + 4.23·31-s − 4.96·33-s + 3.98·35-s − 6.07·37-s − 3.57·39-s − 1.00·41-s + 1.75·43-s − 2.60·45-s − 2.20·47-s − 4.65·49-s + 5.67·51-s + 5.12·53-s − 12.9·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.16·5-s − 0.578·7-s + 0.333·9-s + 1.49·11-s + 0.991·13-s + 0.672·15-s − 1.37·17-s + 1.72·19-s + 0.334·21-s − 1.11·23-s + 0.356·25-s − 0.192·27-s + 0.681·29-s + 0.759·31-s − 0.865·33-s + 0.674·35-s − 0.998·37-s − 0.572·39-s − 0.156·41-s + 0.267·43-s − 0.388·45-s − 0.322·47-s − 0.665·49-s + 0.794·51-s + 0.703·53-s − 1.74·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.123023060\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.123023060\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 503 | \( 1 + T \) |
| good | 5 | \( 1 + 2.60T + 5T^{2} \) |
| 7 | \( 1 + 1.53T + 7T^{2} \) |
| 11 | \( 1 - 4.96T + 11T^{2} \) |
| 13 | \( 1 - 3.57T + 13T^{2} \) |
| 17 | \( 1 + 5.67T + 17T^{2} \) |
| 19 | \( 1 - 7.50T + 19T^{2} \) |
| 23 | \( 1 + 5.35T + 23T^{2} \) |
| 29 | \( 1 - 3.66T + 29T^{2} \) |
| 31 | \( 1 - 4.23T + 31T^{2} \) |
| 37 | \( 1 + 6.07T + 37T^{2} \) |
| 41 | \( 1 + 1.00T + 41T^{2} \) |
| 43 | \( 1 - 1.75T + 43T^{2} \) |
| 47 | \( 1 + 2.20T + 47T^{2} \) |
| 53 | \( 1 - 5.12T + 53T^{2} \) |
| 59 | \( 1 - 1.75T + 59T^{2} \) |
| 61 | \( 1 - 2.51T + 61T^{2} \) |
| 67 | \( 1 - 12.8T + 67T^{2} \) |
| 71 | \( 1 + 7.52T + 71T^{2} \) |
| 73 | \( 1 + 0.588T + 73T^{2} \) |
| 79 | \( 1 - 3.18T + 79T^{2} \) |
| 83 | \( 1 + 1.21T + 83T^{2} \) |
| 89 | \( 1 + 13.4T + 89T^{2} \) |
| 97 | \( 1 + 16.6T + 97T^{2} \) |
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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
−8.139030623217003442761835847243, −7.17600517053596195124399260580, −6.66404265094016103337094321151, −6.13859277493726223242682604130, −5.19537901933541224503614574792, −4.15489448673489971443650555537, −3.89361751899870652599553786417, −3.03392205682969235217423767354, −1.58987345050958878255484150900, −0.60077304548996441154510847183,
0.60077304548996441154510847183, 1.58987345050958878255484150900, 3.03392205682969235217423767354, 3.89361751899870652599553786417, 4.15489448673489971443650555537, 5.19537901933541224503614574792, 6.13859277493726223242682604130, 6.66404265094016103337094321151, 7.17600517053596195124399260580, 8.139030623217003442761835847243
