| L(s) = 1 | + 2-s + 3-s − 4-s − 3·5-s + 6-s + 2·7-s − 3·8-s − 2·9-s − 3·10-s − 3·11-s − 12-s + 13-s + 2·14-s − 3·15-s − 16-s − 2·17-s − 2·18-s − 4·19-s + 3·20-s + 2·21-s − 3·22-s − 4·23-s − 3·24-s + 4·25-s + 26-s − 5·27-s − 2·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s − 1/2·4-s − 1.34·5-s + 0.408·6-s + 0.755·7-s − 1.06·8-s − 2/3·9-s − 0.948·10-s − 0.904·11-s − 0.288·12-s + 0.277·13-s + 0.534·14-s − 0.774·15-s − 1/4·16-s − 0.485·17-s − 0.471·18-s − 0.917·19-s + 0.670·20-s + 0.436·21-s − 0.639·22-s − 0.834·23-s − 0.612·24-s + 4/5·25-s + 0.196·26-s − 0.962·27-s − 0.377·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8033 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8033 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9838230950\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9838230950\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 29 | \( 1 - T \) |
| 277 | \( 1 + T \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 9 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 13 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 3 T + p T^{2} \) |
| 83 | \( 1 - 10 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−8.029742706159624117640755861966, −7.37703461668037548252292425993, −6.29534025377701360004449313040, −5.55633080916977182348973231632, −4.89117299024782644097051405730, −4.10671859923880675297647366568, −3.77871505711256024954745440712, −2.87215146280125656419142596664, −2.10147262351322083877234448628, −0.40418153111815300411771914768,
0.40418153111815300411771914768, 2.10147262351322083877234448628, 2.87215146280125656419142596664, 3.77871505711256024954745440712, 4.10671859923880675297647366568, 4.89117299024782644097051405730, 5.55633080916977182348973231632, 6.29534025377701360004449313040, 7.37703461668037548252292425993, 8.029742706159624117640755861966
