| L(s) = 1 | + 2-s − 3-s + 4-s − 1.69·5-s − 6-s + 3.25·7-s + 8-s + 9-s − 1.69·10-s + 5.34·11-s − 12-s − 0.872·13-s + 3.25·14-s + 1.69·15-s + 16-s + 17-s + 18-s − 8.19·19-s − 1.69·20-s − 3.25·21-s + 5.34·22-s + 6.81·23-s − 24-s − 2.13·25-s − 0.872·26-s − 27-s + 3.25·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.757·5-s − 0.408·6-s + 1.23·7-s + 0.353·8-s + 0.333·9-s − 0.535·10-s + 1.61·11-s − 0.288·12-s − 0.241·13-s + 0.871·14-s + 0.437·15-s + 0.250·16-s + 0.242·17-s + 0.235·18-s − 1.88·19-s − 0.378·20-s − 0.711·21-s + 1.14·22-s + 1.42·23-s − 0.204·24-s − 0.426·25-s − 0.171·26-s − 0.192·27-s + 0.616·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.022885835\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.022885835\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 - T \) |
| good | 5 | \( 1 + 1.69T + 5T^{2} \) |
| 7 | \( 1 - 3.25T + 7T^{2} \) |
| 11 | \( 1 - 5.34T + 11T^{2} \) |
| 13 | \( 1 + 0.872T + 13T^{2} \) |
| 19 | \( 1 + 8.19T + 19T^{2} \) |
| 23 | \( 1 - 6.81T + 23T^{2} \) |
| 29 | \( 1 - 7.06T + 29T^{2} \) |
| 31 | \( 1 - 8.08T + 31T^{2} \) |
| 37 | \( 1 - 5.23T + 37T^{2} \) |
| 41 | \( 1 + 11.8T + 41T^{2} \) |
| 43 | \( 1 - 8.64T + 43T^{2} \) |
| 47 | \( 1 - 2.11T + 47T^{2} \) |
| 53 | \( 1 + 9.00T + 53T^{2} \) |
| 61 | \( 1 - 13.5T + 61T^{2} \) |
| 67 | \( 1 + 8.42T + 67T^{2} \) |
| 71 | \( 1 - 4.49T + 71T^{2} \) |
| 73 | \( 1 + 8.42T + 73T^{2} \) |
| 79 | \( 1 + 4.21T + 79T^{2} \) |
| 83 | \( 1 - 11.0T + 83T^{2} \) |
| 89 | \( 1 + 10.7T + 89T^{2} \) |
| 97 | \( 1 + 0.447T + 97T^{2} \) |
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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
−8.183911124913623970526338304998, −7.09828342580862628564877768542, −6.64936047427448921206807257248, −5.97477520837368297151229326340, −4.95819298576991243846627263390, −4.40763179712554984975412080500, −4.05720806278957139479076397381, −2.90818907448180106344368498673, −1.77536196816128511974458573226, −0.904262550273856531621613818251,
0.904262550273856531621613818251, 1.77536196816128511974458573226, 2.90818907448180106344368498673, 4.05720806278957139479076397381, 4.40763179712554984975412080500, 4.95819298576991243846627263390, 5.97477520837368297151229326340, 6.64936047427448921206807257248, 7.09828342580862628564877768542, 8.183911124913623970526338304998
