| L(s) = 1 | + 2-s + 3-s + 4-s + 3·5-s + 6-s + 8-s + 9-s + 3·10-s + 3·11-s + 12-s + 13-s + 3·15-s + 16-s + 6·17-s + 18-s − 4·19-s + 3·20-s + 3·22-s + 6·23-s + 24-s + 4·25-s + 26-s + 27-s − 9·29-s + 3·30-s + 5·31-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 + 1/3·9-s + 0.948·10-s + 0.904·11-s + 0.288·12-s + 0.277·13-s + 0.774·15-s + 1/4·16-s + 1.45·17-s + 0.235·18-s − 0.917·19-s + 0.670·20-s + 0.639·22-s + 1.25·23-s + 0.204·24-s + 4/5·25-s + 0.196·26-s + 0.192·27-s − 1.67·29-s + 0.547·30-s + 0.898·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.389692051\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.389692051\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 5 T + p T^{2} \) |
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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
−8.531065978282456215457879186909, −7.73356602369668146701739555938, −6.67160146219577668496172268472, −6.38564627606316414560598133391, −5.40543217614005395992672741313, −4.85847338337314529346900720728, −3.65061445423410425106174718394, −3.15341375148222668338048849723, −1.96311823621411457019581167905, −1.40751542574001392118247404333,
1.40751542574001392118247404333, 1.96311823621411457019581167905, 3.15341375148222668338048849723, 3.65061445423410425106174718394, 4.85847338337314529346900720728, 5.40543217614005395992672741313, 6.38564627606316414560598133391, 6.67160146219577668496172268472, 7.73356602369668146701739555938, 8.531065978282456215457879186909
