| L(s) = 1 | − 2.09·2-s + 3-s + 2.38·4-s − 2.09·6-s + 2.59·7-s − 0.809·8-s + 9-s + 2.38·12-s + 6.70·13-s − 5.43·14-s − 3.07·16-s − 5.98·17-s − 2.09·18-s − 2.93·19-s + 2.59·21-s + 1.71·23-s − 0.809·24-s − 14.0·26-s + 27-s + 6.19·28-s − 1.41·29-s + 3.54·31-s + 8.06·32-s + 12.5·34-s + 2.38·36-s − 3.24·37-s + 6.15·38-s + ⋯ |
| L(s) = 1 | − 1.48·2-s + 0.577·3-s + 1.19·4-s − 0.855·6-s + 0.980·7-s − 0.286·8-s + 0.333·9-s + 0.688·12-s + 1.85·13-s − 1.45·14-s − 0.769·16-s − 1.45·17-s − 0.493·18-s − 0.673·19-s + 0.566·21-s + 0.358·23-s − 0.165·24-s − 2.75·26-s + 0.192·27-s + 1.16·28-s − 0.262·29-s + 0.636·31-s + 1.42·32-s + 2.14·34-s + 0.397·36-s − 0.532·37-s + 0.997·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.443037684\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.443037684\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 2.09T + 2T^{2} \) |
| 7 | \( 1 - 2.59T + 7T^{2} \) |
| 13 | \( 1 - 6.70T + 13T^{2} \) |
| 17 | \( 1 + 5.98T + 17T^{2} \) |
| 19 | \( 1 + 2.93T + 19T^{2} \) |
| 23 | \( 1 - 1.71T + 23T^{2} \) |
| 29 | \( 1 + 1.41T + 29T^{2} \) |
| 31 | \( 1 - 3.54T + 31T^{2} \) |
| 37 | \( 1 + 3.24T + 37T^{2} \) |
| 41 | \( 1 - 9.91T + 41T^{2} \) |
| 43 | \( 1 - 0.378T + 43T^{2} \) |
| 47 | \( 1 + 5.21T + 47T^{2} \) |
| 53 | \( 1 + 2.34T + 53T^{2} \) |
| 59 | \( 1 + 11.8T + 59T^{2} \) |
| 61 | \( 1 - 12.1T + 61T^{2} \) |
| 67 | \( 1 + 8.51T + 67T^{2} \) |
| 71 | \( 1 + 6.97T + 71T^{2} \) |
| 73 | \( 1 - 2.53T + 73T^{2} \) |
| 79 | \( 1 - 9.29T + 79T^{2} \) |
| 83 | \( 1 + 8.94T + 83T^{2} \) |
| 89 | \( 1 - 9.83T + 89T^{2} \) |
| 97 | \( 1 - 17.0T + 97T^{2} \) |
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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
−7.978547692462159927733997223225, −7.37939950531993478829551447197, −6.54905448988125469536967930954, −6.04601026855282530979554286547, −4.73815635886961390742887950218, −4.28088346491779132565429096530, −3.27380954683923651150476246851, −2.15879582489636971786268110343, −1.65371441408447097164172658953, −0.74036742042948559085449863843,
0.74036742042948559085449863843, 1.65371441408447097164172658953, 2.15879582489636971786268110343, 3.27380954683923651150476246851, 4.28088346491779132565429096530, 4.73815635886961390742887950218, 6.04601026855282530979554286547, 6.54905448988125469536967930954, 7.37939950531993478829551447197, 7.978547692462159927733997223225
