| L(s) = 1 | − 0.473·2-s − 1.77·4-s + 2.56·7-s + 1.78·8-s + 6.16·11-s + 2.13·13-s − 1.21·14-s + 2.70·16-s − 3.16·17-s + 0.356·19-s − 2.91·22-s + 4.21·23-s − 1.00·26-s − 4.55·28-s − 1.68·29-s − 8.25·31-s − 4.85·32-s + 1.49·34-s + 3.63·37-s − 0.168·38-s + 2.73·41-s − 7.67·43-s − 10.9·44-s − 1.99·46-s + 11.4·47-s − 0.430·49-s − 3.78·52-s + ⋯ |
| L(s) = 1 | − 0.334·2-s − 0.888·4-s + 0.968·7-s + 0.631·8-s + 1.85·11-s + 0.591·13-s − 0.324·14-s + 0.676·16-s − 0.768·17-s + 0.0817·19-s − 0.622·22-s + 0.878·23-s − 0.197·26-s − 0.860·28-s − 0.313·29-s − 1.48·31-s − 0.858·32-s + 0.257·34-s + 0.597·37-s − 0.0273·38-s + 0.426·41-s − 1.17·43-s − 1.65·44-s − 0.293·46-s + 1.66·47-s − 0.0615·49-s − 0.525·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.551994584\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.551994584\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 0.473T + 2T^{2} \) |
| 7 | \( 1 - 2.56T + 7T^{2} \) |
| 11 | \( 1 - 6.16T + 11T^{2} \) |
| 13 | \( 1 - 2.13T + 13T^{2} \) |
| 17 | \( 1 + 3.16T + 17T^{2} \) |
| 19 | \( 1 - 0.356T + 19T^{2} \) |
| 23 | \( 1 - 4.21T + 23T^{2} \) |
| 29 | \( 1 + 1.68T + 29T^{2} \) |
| 31 | \( 1 + 8.25T + 31T^{2} \) |
| 37 | \( 1 - 3.63T + 37T^{2} \) |
| 41 | \( 1 - 2.73T + 41T^{2} \) |
| 43 | \( 1 + 7.67T + 43T^{2} \) |
| 47 | \( 1 - 11.4T + 47T^{2} \) |
| 53 | \( 1 - 9.43T + 53T^{2} \) |
| 59 | \( 1 + 10.2T + 59T^{2} \) |
| 61 | \( 1 + 0.0109T + 61T^{2} \) |
| 67 | \( 1 - 0.982T + 67T^{2} \) |
| 71 | \( 1 - 6.43T + 71T^{2} \) |
| 73 | \( 1 - 6.61T + 73T^{2} \) |
| 79 | \( 1 + 9.47T + 79T^{2} \) |
| 83 | \( 1 - 10.4T + 83T^{2} \) |
| 89 | \( 1 - 6.26T + 89T^{2} \) |
| 97 | \( 1 - 7.20T + 97T^{2} \) |
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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
−8.978379663275627300551125814977, −8.684946806056705954170088344659, −7.69446484465043850003849474973, −6.91537778450797389228322178279, −5.94612734004827364700429546065, −4.99589375078119865612437326590, −4.21455659274653560642816607563, −3.58278698217385449218391801847, −1.85029490493925111992155641512, −0.962506383000096193492673335512,
0.962506383000096193492673335512, 1.85029490493925111992155641512, 3.58278698217385449218391801847, 4.21455659274653560642816607563, 4.99589375078119865612437326590, 5.94612734004827364700429546065, 6.91537778450797389228322178279, 7.69446484465043850003849474973, 8.684946806056705954170088344659, 8.978379663275627300551125814977
