L(s) = 1 | − 2-s + 3-s + 4-s + 4·5-s − 6-s − 7-s − 8-s − 2·9-s − 4·10-s − 11-s + 12-s + 13-s + 14-s + 4·15-s + 16-s + 4·17-s + 2·18-s + 2·19-s + 4·20-s − 21-s + 22-s − 7·23-s − 24-s + 11·25-s − 26-s − 5·27-s − 28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 1.78·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s − 2/3·9-s − 1.26·10-s − 0.301·11-s + 0.288·12-s + 0.277·13-s + 0.267·14-s + 1.03·15-s + 1/4·16-s + 0.970·17-s + 0.471·18-s + 0.458·19-s + 0.894·20-s − 0.218·21-s + 0.213·22-s − 1.45·23-s − 0.204·24-s + 11/5·25-s − 0.196·26-s − 0.962·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 182 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 182 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.208975634\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.208975634\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 - T \) |
good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 9 T + p T^{2} \) |
| 79 | \( 1 + 13 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 11 T + p T^{2} \) |
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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
−12.80366415530887141712878791219, −11.49145261462399052030547351428, −10.16850461541097430076578447843, −9.706455892648275971763122964510, −8.805211576902181942047270874289, −7.73089602080583279532021184972, −6.23512381121606234073815226068, −5.54572200207102921430114601568, −3.13073144396076657090075619119, −1.89940553571645055742521325526,
1.89940553571645055742521325526, 3.13073144396076657090075619119, 5.54572200207102921430114601568, 6.23512381121606234073815226068, 7.73089602080583279532021184972, 8.805211576902181942047270874289, 9.706455892648275971763122964510, 10.16850461541097430076578447843, 11.49145261462399052030547351428, 12.80366415530887141712878791219