L(s) = 1 | − 2-s + 2·3-s + 4-s + 5-s − 2·6-s + 7-s − 8-s + 9-s − 10-s + 3·11-s + 2·12-s − 14-s + 2·15-s + 16-s + 3·17-s − 18-s − 6·19-s + 20-s + 2·21-s − 3·22-s + 2·23-s − 2·24-s + 25-s − 4·27-s + 28-s − 3·29-s − 2·30-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.15·3-s + 1/2·4-s + 0.447·5-s − 0.816·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.904·11-s + 0.577·12-s − 0.267·14-s + 0.516·15-s + 1/4·16-s + 0.727·17-s − 0.235·18-s − 1.37·19-s + 0.223·20-s + 0.436·21-s − 0.639·22-s + 0.417·23-s − 0.408·24-s + 1/5·25-s − 0.769·27-s + 0.188·28-s − 0.557·29-s − 0.365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.579927740\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.579927740\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 37 | \( 1 + T \) |
good | 3 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 13 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 15 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 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
−11.22168018362876670458435813477, −10.25012054118373965775786817728, −9.320554490798598393846716062676, −8.720845365035222845099208301732, −7.947480595907964918208604648618, −6.89931657836489782864864420449, −5.77272193873314481326296907642, −4.12586421650220620613319696777, −2.82505147340032154850850367219, −1.63805431218632083963465992264,
1.63805431218632083963465992264, 2.82505147340032154850850367219, 4.12586421650220620613319696777, 5.77272193873314481326296907642, 6.89931657836489782864864420449, 7.947480595907964918208604648618, 8.720845365035222845099208301732, 9.320554490798598393846716062676, 10.25012054118373965775786817728, 11.22168018362876670458435813477