| L(s) = 1 | + 3.11·2-s + 0.0197·3-s + 1.71·4-s − 5·5-s + 0.0616·6-s − 19.0·7-s − 19.5·8-s − 26.9·9-s − 15.5·10-s + 11·11-s + 0.0340·12-s − 32.2·13-s − 59.3·14-s − 0.0989·15-s − 74.7·16-s + 51.1·17-s − 84.1·18-s + 19·19-s − 8.59·20-s − 0.377·21-s + 34.2·22-s + 106.·23-s − 0.387·24-s + 25·25-s − 100.·26-s − 1.06·27-s − 32.7·28-s + ⋯ |
| L(s) = 1 | + 1.10·2-s + 0.00380·3-s + 0.214·4-s − 0.447·5-s + 0.00419·6-s − 1.02·7-s − 0.865·8-s − 0.999·9-s − 0.492·10-s + 0.301·11-s + 0.000818·12-s − 0.688·13-s − 1.13·14-s − 0.00170·15-s − 1.16·16-s + 0.730·17-s − 1.10·18-s + 0.229·19-s − 0.0960·20-s − 0.00391·21-s + 0.332·22-s + 0.965·23-s − 0.00329·24-s + 0.200·25-s − 0.759·26-s − 0.00761·27-s − 0.221·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.799004099\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.799004099\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 5T \) |
| 11 | \( 1 - 11T \) |
| 19 | \( 1 - 19T \) |
| good | 2 | \( 1 - 3.11T + 8T^{2} \) |
| 3 | \( 1 - 0.0197T + 27T^{2} \) |
| 7 | \( 1 + 19.0T + 343T^{2} \) |
| 13 | \( 1 + 32.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 51.1T + 4.91e3T^{2} \) |
| 23 | \( 1 - 106.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 56.5T + 2.43e4T^{2} \) |
| 31 | \( 1 - 281.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 97.5T + 5.06e4T^{2} \) |
| 41 | \( 1 - 227.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 97.9T + 7.95e4T^{2} \) |
| 47 | \( 1 - 100.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 660.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 868.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 0.00261T + 2.26e5T^{2} \) |
| 67 | \( 1 + 142.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 369.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 72.0T + 3.89e5T^{2} \) |
| 79 | \( 1 - 152.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 133.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 533.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 92.7T + 9.12e5T^{2} \) |
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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
−9.419907819487775154973596710694, −8.860042892104506410529511622579, −7.75980491238629179586369948266, −6.74252596418403471745106609077, −5.96605739365968558485953720758, −5.19338439476448404195051053392, −4.23537609625603114189957457606, −3.23193466641353633004404669927, −2.74120394497051166026112670297, −0.57429393122080200537658921937,
0.57429393122080200537658921937, 2.74120394497051166026112670297, 3.23193466641353633004404669927, 4.23537609625603114189957457606, 5.19338439476448404195051053392, 5.96605739365968558485953720758, 6.74252596418403471745106609077, 7.75980491238629179586369948266, 8.860042892104506410529511622579, 9.419907819487775154973596710694
