| L(s) = 1 | − 3·3-s − 4·5-s + 4·7-s + 9·9-s − 2·11-s + 13·13-s + 12·15-s − 6·17-s + 36·19-s − 12·21-s − 20·23-s − 109·25-s − 27·27-s + 14·29-s − 152·31-s + 6·33-s − 16·35-s + 258·37-s − 39·39-s + 84·41-s + 188·43-s − 36·45-s + 254·47-s − 327·49-s + 18·51-s − 366·53-s + 8·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.357·5-s + 0.215·7-s + 1/3·9-s − 0.0548·11-s + 0.277·13-s + 0.206·15-s − 0.0856·17-s + 0.434·19-s − 0.124·21-s − 0.181·23-s − 0.871·25-s − 0.192·27-s + 0.0896·29-s − 0.880·31-s + 0.0316·33-s − 0.0772·35-s + 1.14·37-s − 0.160·39-s + 0.319·41-s + 0.666·43-s − 0.119·45-s + 0.788·47-s − 0.953·49-s + 0.0494·51-s − 0.948·53-s + 0.0196·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p T \) |
| 13 | \( 1 - p T \) |
| good | 5 | \( 1 + 4 T + p^{3} T^{2} \) |
| 7 | \( 1 - 4 T + p^{3} T^{2} \) |
| 11 | \( 1 + 2 T + p^{3} T^{2} \) |
| 17 | \( 1 + 6 T + p^{3} T^{2} \) |
| 19 | \( 1 - 36 T + p^{3} T^{2} \) |
| 23 | \( 1 + 20 T + p^{3} T^{2} \) |
| 29 | \( 1 - 14 T + p^{3} T^{2} \) |
| 31 | \( 1 + 152 T + p^{3} T^{2} \) |
| 37 | \( 1 - 258 T + p^{3} T^{2} \) |
| 41 | \( 1 - 84 T + p^{3} T^{2} \) |
| 43 | \( 1 - 188 T + p^{3} T^{2} \) |
| 47 | \( 1 - 254 T + p^{3} T^{2} \) |
| 53 | \( 1 + 366 T + p^{3} T^{2} \) |
| 59 | \( 1 + 550 T + p^{3} T^{2} \) |
| 61 | \( 1 - 14 T + p^{3} T^{2} \) |
| 67 | \( 1 + 448 T + p^{3} T^{2} \) |
| 71 | \( 1 - 926 T + p^{3} T^{2} \) |
| 73 | \( 1 - 254 T + p^{3} T^{2} \) |
| 79 | \( 1 - 1328 T + p^{3} T^{2} \) |
| 83 | \( 1 + 186 T + p^{3} T^{2} \) |
| 89 | \( 1 + 336 T + p^{3} T^{2} \) |
| 97 | \( 1 - 614 T + p^{3} T^{2} \) |
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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
−7.912366220723890369516573695271, −7.61581933790446180252836238283, −6.53507486450846083956414278581, −5.89201528917905983468462258051, −5.04963136017754377321614603052, −4.23002385721650404749144444605, −3.41215187259906838581463819489, −2.19226093362211149035204977251, −1.08961867194871394276180834490, 0,
1.08961867194871394276180834490, 2.19226093362211149035204977251, 3.41215187259906838581463819489, 4.23002385721650404749144444605, 5.04963136017754377321614603052, 5.89201528917905983468462258051, 6.53507486450846083956414278581, 7.61581933790446180252836238283, 7.912366220723890369516573695271
