| L(s) = 1 | + 2-s + 3-s + 4-s + 3·5-s + 6-s + 8-s − 2·9-s + 3·10-s − 11-s + 12-s + 13-s + 3·15-s + 16-s + 4·17-s − 2·18-s + 4·19-s + 3·20-s − 22-s − 2·23-s + 24-s + 4·25-s + 26-s − 5·27-s + 29-s + 3·30-s + 31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 1.34·5-s + 0.408·6-s + 0.353·8-s − 2/3·9-s + 0.948·10-s − 0.301·11-s + 0.288·12-s + 0.277·13-s + 0.774·15-s + 1/4·16-s + 0.970·17-s − 0.471·18-s + 0.917·19-s + 0.670·20-s − 0.213·22-s − 0.417·23-s + 0.204·24-s + 4/5·25-s + 0.196·26-s − 0.962·27-s + 0.185·29-s + 0.547·30-s + 0.179·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2842 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2842 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.541378955\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.541378955\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 3 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + 13 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−8.853314417459682240676482464926, −7.958154421944426163012185389075, −7.29855643233325696806523744698, −6.11848672881495667406425312278, −5.77883523579558920548200346517, −5.08744054037696137933920918348, −3.92472525937793828828847236707, −2.95969743538279746654139728606, −2.40223994180230416940623783145, −1.28138267371571222784089274599,
1.28138267371571222784089274599, 2.40223994180230416940623783145, 2.95969743538279746654139728606, 3.92472525937793828828847236707, 5.08744054037696137933920918348, 5.77883523579558920548200346517, 6.11848672881495667406425312278, 7.29855643233325696806523744698, 7.958154421944426163012185389075, 8.853314417459682240676482464926
