| L(s) = 1 | − 3-s + 2·5-s + 4·7-s − 2·9-s + 2·11-s + 7·13-s − 2·15-s + 4·17-s + 6·19-s − 4·21-s − 25-s + 5·27-s + 5·29-s + 3·31-s − 2·33-s + 8·35-s − 2·37-s − 7·39-s − 9·41-s − 8·43-s − 4·45-s − 47-s + 9·49-s − 4·51-s + 6·53-s + 4·55-s − 6·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s + 1.51·7-s − 2/3·9-s + 0.603·11-s + 1.94·13-s − 0.516·15-s + 0.970·17-s + 1.37·19-s − 0.872·21-s − 1/5·25-s + 0.962·27-s + 0.928·29-s + 0.538·31-s − 0.348·33-s + 1.35·35-s − 0.328·37-s − 1.12·39-s − 1.40·41-s − 1.21·43-s − 0.596·45-s − 0.145·47-s + 9/7·49-s − 0.560·51-s + 0.824·53-s + 0.539·55-s − 0.794·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.753284331\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.753284331\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 7 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 13 T + p T^{2} \) |
| 73 | \( 1 + 3 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 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
−8.463523181421092226706669526699, −7.82294576929983536521825169596, −6.72912342512596841517948904434, −6.05220492851837615274997431163, −5.45258791200978400763978660919, −4.96279447908039424391057317925, −3.83077416135681826938358923048, −2.94668432293515666131888787217, −1.53923462014814349403579065440, −1.17461094001747730469015874270,
1.17461094001747730469015874270, 1.53923462014814349403579065440, 2.94668432293515666131888787217, 3.83077416135681826938358923048, 4.96279447908039424391057317925, 5.45258791200978400763978660919, 6.05220492851837615274997431163, 6.72912342512596841517948904434, 7.82294576929983536521825169596, 8.463523181421092226706669526699
