| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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
−9.879508395072031981848342443154, −9.058322193075846324997380625345, −7.921959058161173120552124514900, −6.99261477049534038789432211053, −6.08799657551806535381486001574, −5.26543769838045674371225081095, −4.21855753050061565903296679334, −2.89566013504808123096798754099, −1.50537601340230927533980421494, 0,
1.50537601340230927533980421494, 2.89566013504808123096798754099, 4.21855753050061565903296679334, 5.26543769838045674371225081095, 6.08799657551806535381486001574, 6.99261477049534038789432211053, 7.921959058161173120552124514900, 9.058322193075846324997380625345, 9.879508395072031981848342443154
