| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 3·7-s − 8-s + 9-s − 11-s − 12-s + 3·14-s + 16-s − 17-s − 18-s − 6·19-s + 3·21-s + 22-s + 24-s − 5·25-s − 27-s − 3·28-s + 6·29-s − 32-s + 33-s + 34-s + 36-s + 3·37-s + 6·38-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 1.13·7-s − 0.353·8-s + 1/3·9-s − 0.301·11-s − 0.288·12-s + 0.801·14-s + 1/4·16-s − 0.242·17-s − 0.235·18-s − 1.37·19-s + 0.654·21-s + 0.213·22-s + 0.204·24-s − 25-s − 0.192·27-s − 0.566·28-s + 1.11·29-s − 0.176·32-s + 0.174·33-s + 0.171·34-s + 1/6·36-s + 0.493·37-s + 0.973·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189618 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189618 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8486414983\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8486414983\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−12.91707510960277, −12.65940122009894, −11.99597021620348, −11.87877910495464, −11.01373508685144, −10.57586219213828, −10.50402337554713, −9.675637003808099, −9.492144115934789, −8.919132569685527, −8.336394599132650, −7.917617252050303, −7.269940853080808, −6.814800075219978, −6.402487733057667, −5.859514856232382, −5.614764648306863, −4.681659930191286, −4.148549003688299, −3.736147400077463, −2.805005116893289, −2.484400680790480, −1.819196930663966, −0.8534091206802280, −0.3941937048398590,
0.3941937048398590, 0.8534091206802280, 1.819196930663966, 2.484400680790480, 2.805005116893289, 3.736147400077463, 4.148549003688299, 4.681659930191286, 5.614764648306863, 5.859514856232382, 6.402487733057667, 6.814800075219978, 7.269940853080808, 7.917617252050303, 8.336394599132650, 8.919132569685527, 9.492144115934789, 9.675637003808099, 10.50402337554713, 10.57586219213828, 11.01373508685144, 11.87877910495464, 11.99597021620348, 12.65940122009894, 12.91707510960277
