| L(s) = 1 | − 0.329·2-s − 1.89·4-s − 2.26·7-s + 1.28·8-s + 1.59·11-s + 3.85·13-s + 0.746·14-s + 3.35·16-s − 17-s − 2.50·19-s − 0.524·22-s − 0.0656·23-s − 1.27·26-s + 4.27·28-s − 9.23·29-s + 2.34·31-s − 3.67·32-s + 0.329·34-s + 4.48·37-s + 0.825·38-s + 0.635·41-s − 4.07·43-s − 3.00·44-s + 0.0216·46-s + 2.43·47-s − 1.88·49-s − 7.28·52-s + ⋯ |
| L(s) = 1 | − 0.233·2-s − 0.945·4-s − 0.855·7-s + 0.453·8-s + 0.479·11-s + 1.06·13-s + 0.199·14-s + 0.839·16-s − 0.242·17-s − 0.574·19-s − 0.111·22-s − 0.0136·23-s − 0.249·26-s + 0.808·28-s − 1.71·29-s + 0.421·31-s − 0.649·32-s + 0.0565·34-s + 0.737·37-s + 0.133·38-s + 0.0992·41-s − 0.621·43-s − 0.453·44-s + 0.00319·46-s + 0.355·47-s − 0.268·49-s − 1.01·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.007410752\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.007410752\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 2 | \( 1 + 0.329T + 2T^{2} \) |
| 7 | \( 1 + 2.26T + 7T^{2} \) |
| 11 | \( 1 - 1.59T + 11T^{2} \) |
| 13 | \( 1 - 3.85T + 13T^{2} \) |
| 19 | \( 1 + 2.50T + 19T^{2} \) |
| 23 | \( 1 + 0.0656T + 23T^{2} \) |
| 29 | \( 1 + 9.23T + 29T^{2} \) |
| 31 | \( 1 - 2.34T + 31T^{2} \) |
| 37 | \( 1 - 4.48T + 37T^{2} \) |
| 41 | \( 1 - 0.635T + 41T^{2} \) |
| 43 | \( 1 + 4.07T + 43T^{2} \) |
| 47 | \( 1 - 2.43T + 47T^{2} \) |
| 53 | \( 1 - 3.09T + 53T^{2} \) |
| 59 | \( 1 - 13.7T + 59T^{2} \) |
| 61 | \( 1 + 1.22T + 61T^{2} \) |
| 67 | \( 1 + 9.19T + 67T^{2} \) |
| 71 | \( 1 + 5.79T + 71T^{2} \) |
| 73 | \( 1 + 10.2T + 73T^{2} \) |
| 79 | \( 1 + 5.88T + 79T^{2} \) |
| 83 | \( 1 - 15.5T + 83T^{2} \) |
| 89 | \( 1 - 0.221T + 89T^{2} \) |
| 97 | \( 1 - 14.6T + 97T^{2} \) |
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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
−8.728971521115228914775020998143, −7.88616315062681000112958748297, −7.04545717735623835856241565376, −6.18038449939613986117386034951, −5.65269004350018145854745185935, −4.53893883638924190586974052511, −3.88097367040457493019378144209, −3.20726152587279443102433722252, −1.81433296368296894766285372251, −0.61270964215936419525305071891,
0.61270964215936419525305071891, 1.81433296368296894766285372251, 3.20726152587279443102433722252, 3.88097367040457493019378144209, 4.53893883638924190586974052511, 5.65269004350018145854745185935, 6.18038449939613986117386034951, 7.04545717735623835856241565376, 7.88616315062681000112958748297, 8.728971521115228914775020998143
