| L(s) = 1 | − 2.71·3-s − 0.0156·7-s + 4.35·9-s + 6.41·11-s − 2.76·13-s − 2.07·17-s + 4.25·19-s + 0.0424·21-s + 5.97·23-s − 3.67·27-s + 5.37·29-s − 3.59·31-s − 17.3·33-s + 9.04·37-s + 7.48·39-s − 7.03·41-s − 4.96·43-s + 5.73·47-s − 6.99·49-s + 5.63·51-s − 10.5·53-s − 11.5·57-s + 1.13·59-s + 8.76·61-s − 0.0682·63-s + 11.0·67-s − 16.2·69-s + ⋯ |
| L(s) = 1 | − 1.56·3-s − 0.00591·7-s + 1.45·9-s + 1.93·11-s − 0.765·13-s − 0.503·17-s + 0.976·19-s + 0.00926·21-s + 1.24·23-s − 0.707·27-s + 0.998·29-s − 0.645·31-s − 3.02·33-s + 1.48·37-s + 1.19·39-s − 1.09·41-s − 0.756·43-s + 0.836·47-s − 0.999·49-s + 0.788·51-s − 1.45·53-s − 1.52·57-s + 0.147·59-s + 1.12·61-s − 0.00859·63-s + 1.34·67-s − 1.95·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.174163189\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.174163189\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 2.71T + 3T^{2} \) |
| 7 | \( 1 + 0.0156T + 7T^{2} \) |
| 11 | \( 1 - 6.41T + 11T^{2} \) |
| 13 | \( 1 + 2.76T + 13T^{2} \) |
| 17 | \( 1 + 2.07T + 17T^{2} \) |
| 19 | \( 1 - 4.25T + 19T^{2} \) |
| 23 | \( 1 - 5.97T + 23T^{2} \) |
| 29 | \( 1 - 5.37T + 29T^{2} \) |
| 31 | \( 1 + 3.59T + 31T^{2} \) |
| 37 | \( 1 - 9.04T + 37T^{2} \) |
| 41 | \( 1 + 7.03T + 41T^{2} \) |
| 43 | \( 1 + 4.96T + 43T^{2} \) |
| 47 | \( 1 - 5.73T + 47T^{2} \) |
| 53 | \( 1 + 10.5T + 53T^{2} \) |
| 59 | \( 1 - 1.13T + 59T^{2} \) |
| 61 | \( 1 - 8.76T + 61T^{2} \) |
| 67 | \( 1 - 11.0T + 67T^{2} \) |
| 71 | \( 1 + 11.0T + 71T^{2} \) |
| 73 | \( 1 + 7.38T + 73T^{2} \) |
| 79 | \( 1 - 13.4T + 79T^{2} \) |
| 83 | \( 1 - 3.44T + 83T^{2} \) |
| 89 | \( 1 + 9.49T + 89T^{2} \) |
| 97 | \( 1 - 9.21T + 97T^{2} \) |
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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
−8.516081620375125130109334771783, −7.40233536277396324035988824057, −6.67999610935587256834527495147, −6.43698634337159075224958741171, −5.42244116753834411195316403958, −4.82942916001445891427704759402, −4.12030512100496072234391150045, −3.05085770675102854732665702675, −1.57103225844582139590880022156, −0.72606013535257107177248168773,
0.72606013535257107177248168773, 1.57103225844582139590880022156, 3.05085770675102854732665702675, 4.12030512100496072234391150045, 4.82942916001445891427704759402, 5.42244116753834411195316403958, 6.43698634337159075224958741171, 6.67999610935587256834527495147, 7.40233536277396324035988824057, 8.516081620375125130109334771783
