| L(s) = 1 | + 3·3-s + 12·7-s + 9·9-s − 60·11-s + 42·13-s − 10·17-s − 132·19-s + 36·21-s − 48·23-s + 27·27-s + 226·29-s + 252·31-s − 180·33-s + 362·37-s + 126·39-s − 94·41-s − 228·43-s − 408·47-s − 199·49-s − 30·51-s − 346·53-s − 396·57-s + 300·59-s − 466·61-s + 108·63-s + 204·67-s − 144·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.647·7-s + 1/3·9-s − 1.64·11-s + 0.896·13-s − 0.142·17-s − 1.59·19-s + 0.374·21-s − 0.435·23-s + 0.192·27-s + 1.44·29-s + 1.46·31-s − 0.949·33-s + 1.60·37-s + 0.517·39-s − 0.358·41-s − 0.808·43-s − 1.26·47-s − 0.580·49-s − 0.0823·51-s − 0.896·53-s − 0.920·57-s + 0.661·59-s − 0.978·61-s + 0.215·63-s + 0.371·67-s − 0.251·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - p T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 12 T + p^{3} T^{2} \) |
| 11 | \( 1 + 60 T + p^{3} T^{2} \) |
| 13 | \( 1 - 42 T + p^{3} T^{2} \) |
| 17 | \( 1 + 10 T + p^{3} T^{2} \) |
| 19 | \( 1 + 132 T + p^{3} T^{2} \) |
| 23 | \( 1 + 48 T + p^{3} T^{2} \) |
| 29 | \( 1 - 226 T + p^{3} T^{2} \) |
| 31 | \( 1 - 252 T + p^{3} T^{2} \) |
| 37 | \( 1 - 362 T + p^{3} T^{2} \) |
| 41 | \( 1 + 94 T + p^{3} T^{2} \) |
| 43 | \( 1 + 228 T + p^{3} T^{2} \) |
| 47 | \( 1 + 408 T + p^{3} T^{2} \) |
| 53 | \( 1 + 346 T + p^{3} T^{2} \) |
| 59 | \( 1 - 300 T + p^{3} T^{2} \) |
| 61 | \( 1 + 466 T + p^{3} T^{2} \) |
| 67 | \( 1 - 204 T + p^{3} T^{2} \) |
| 71 | \( 1 + 1056 T + p^{3} T^{2} \) |
| 73 | \( 1 + 330 T + p^{3} T^{2} \) |
| 79 | \( 1 + 612 T + p^{3} T^{2} \) |
| 83 | \( 1 - 564 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1510 T + p^{3} T^{2} \) |
| 97 | \( 1 + 594 T + p^{3} T^{2} \) |
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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
−8.256468438831887933781403917536, −7.81532579164849954207019493127, −6.62828748493883602962271958266, −5.99244656671078955183874261983, −4.80300716668644635040143904581, −4.37932943854135424414553958471, −3.09396995671534425413994647435, −2.40267584280155468072173018525, −1.37152368280162061285975421973, 0,
1.37152368280162061285975421973, 2.40267584280155468072173018525, 3.09396995671534425413994647435, 4.37932943854135424414553958471, 4.80300716668644635040143904581, 5.99244656671078955183874261983, 6.62828748493883602962271958266, 7.81532579164849954207019493127, 8.256468438831887933781403917536
