| L(s) = 1 | + 2-s + 3-s + 4-s + 3·5-s + 6-s − 7-s + 8-s + 9-s + 3·10-s + 4·11-s + 12-s − 3·13-s − 14-s + 3·15-s + 16-s − 4·17-s + 18-s + 3·20-s − 21-s + 4·22-s + 23-s + 24-s + 4·25-s − 3·26-s + 27-s − 28-s + 3·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 1.34·5-s + 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.948·10-s + 1.20·11-s + 0.288·12-s − 0.832·13-s − 0.267·14-s + 0.774·15-s + 1/4·16-s − 0.970·17-s + 0.235·18-s + 0.670·20-s − 0.218·21-s + 0.852·22-s + 0.208·23-s + 0.204·24-s + 4/5·25-s − 0.588·26-s + 0.192·27-s − 0.188·28-s + 0.557·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.593527008\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.593527008\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 9 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 + 3 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−9.843827071515005252615180187856, −9.345059082219016767573214421285, −8.533329024037475026270064025246, −7.11624966976560567011680890410, −6.61433587019532483612650095921, −5.70303300160941996822497347087, −4.72846979566270558944303333386, −3.67780852772662808140519843755, −2.54506917140467324126940654920, −1.68100316040744687359155389371,
1.68100316040744687359155389371, 2.54506917140467324126940654920, 3.67780852772662808140519843755, 4.72846979566270558944303333386, 5.70303300160941996822497347087, 6.61433587019532483612650095921, 7.11624966976560567011680890410, 8.533329024037475026270064025246, 9.345059082219016767573214421285, 9.843827071515005252615180187856
