L(s) = 1 | − 2-s − 3-s + 4-s − 4·5-s + 6-s − 2·7-s − 8-s + 9-s + 4·10-s − 2·11-s − 12-s + 2·13-s + 2·14-s + 4·15-s + 16-s + 17-s − 18-s − 4·19-s − 4·20-s + 2·21-s + 2·22-s − 8·23-s + 24-s + 11·25-s − 2·26-s − 27-s − 2·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 1.78·5-s + 0.408·6-s − 0.755·7-s − 0.353·8-s + 1/3·9-s + 1.26·10-s − 0.603·11-s − 0.288·12-s + 0.554·13-s + 0.534·14-s + 1.03·15-s + 1/4·16-s + 0.242·17-s − 0.235·18-s − 0.917·19-s − 0.894·20-s + 0.436·21-s + 0.426·22-s − 1.66·23-s + 0.204·24-s + 11/5·25-s − 0.392·26-s − 0.192·27-s − 0.377·28-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
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−7.44926473495736617730973627289, −6.78242945869933390685890829954, −6.18298735106427882300982655837, −5.26577183799875089963859955492, −4.24947579665366933702320844086, −3.69182376919432480985940430844, −2.89038182419910934885912913045, −1.56122715193216550988133098271, 0, 0,
1.56122715193216550988133098271, 2.89038182419910934885912913045, 3.69182376919432480985940430844, 4.24947579665366933702320844086, 5.26577183799875089963859955492, 6.18298735106427882300982655837, 6.78242945869933390685890829954, 7.44926473495736617730973627289