| L(s) = 1 | − 3-s − 3·5-s + 3·7-s + 9-s − 3·11-s + 3·15-s + 17-s − 19-s − 3·21-s + 4·25-s − 27-s + 8·29-s − 2·31-s + 3·33-s − 9·35-s + 4·37-s − 12·41-s + 43-s − 3·45-s + 9·47-s + 2·49-s − 51-s − 6·53-s + 9·55-s + 57-s + 6·59-s + 61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.34·5-s + 1.13·7-s + 1/3·9-s − 0.904·11-s + 0.774·15-s + 0.242·17-s − 0.229·19-s − 0.654·21-s + 4/5·25-s − 0.192·27-s + 1.48·29-s − 0.359·31-s + 0.522·33-s − 1.52·35-s + 0.657·37-s − 1.87·41-s + 0.152·43-s − 0.447·45-s + 1.31·47-s + 2/7·49-s − 0.140·51-s − 0.824·53-s + 1.21·55-s + 0.132·57-s + 0.781·59-s + 0.128·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−8.237624252527661573002760073002, −7.48793073157026184368792859893, −6.87628373540077533187937919721, −5.79748059767373498933868247063, −4.96956909965823360434375595852, −4.49729828391185666314372294494, −3.62101959506557558609426499807, −2.53017421160268257135748293220, −1.22959333107214166026800767199, 0,
1.22959333107214166026800767199, 2.53017421160268257135748293220, 3.62101959506557558609426499807, 4.49729828391185666314372294494, 4.96956909965823360434375595852, 5.79748059767373498933868247063, 6.87628373540077533187937919721, 7.48793073157026184368792859893, 8.237624252527661573002760073002
