| L(s) = 1 | + 2-s + 1.73·3-s + 4-s + 5-s + 1.73·6-s − 4.26·7-s + 8-s − 0.00424·9-s + 10-s + 11-s + 1.73·12-s − 4.38·13-s − 4.26·14-s + 1.73·15-s + 16-s − 1.50·17-s − 0.00424·18-s + 5.56·19-s + 20-s − 7.38·21-s + 22-s + 7.10·23-s + 1.73·24-s + 25-s − 4.38·26-s − 5.19·27-s − 4.26·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.999·3-s + 0.5·4-s + 0.447·5-s + 0.706·6-s − 1.61·7-s + 0.353·8-s − 0.00141·9-s + 0.316·10-s + 0.301·11-s + 0.499·12-s − 1.21·13-s − 1.13·14-s + 0.446·15-s + 0.250·16-s − 0.364·17-s − 0.00100·18-s + 1.27·19-s + 0.223·20-s − 1.61·21-s + 0.213·22-s + 1.48·23-s + 0.353·24-s + 0.200·25-s − 0.859·26-s − 1.00·27-s − 0.805·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.077549200\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.077549200\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 73 | \( 1 + T \) |
| good | 3 | \( 1 - 1.73T + 3T^{2} \) |
| 7 | \( 1 + 4.26T + 7T^{2} \) |
| 13 | \( 1 + 4.38T + 13T^{2} \) |
| 17 | \( 1 + 1.50T + 17T^{2} \) |
| 19 | \( 1 - 5.56T + 19T^{2} \) |
| 23 | \( 1 - 7.10T + 23T^{2} \) |
| 29 | \( 1 + 3.43T + 29T^{2} \) |
| 31 | \( 1 + 1.50T + 31T^{2} \) |
| 37 | \( 1 - 10.6T + 37T^{2} \) |
| 41 | \( 1 - 7.93T + 41T^{2} \) |
| 43 | \( 1 + 5.23T + 43T^{2} \) |
| 47 | \( 1 - 8.79T + 47T^{2} \) |
| 53 | \( 1 - 9.80T + 53T^{2} \) |
| 59 | \( 1 - 8.09T + 59T^{2} \) |
| 61 | \( 1 - 7.08T + 61T^{2} \) |
| 67 | \( 1 - 14.5T + 67T^{2} \) |
| 71 | \( 1 + 3.19T + 71T^{2} \) |
| 79 | \( 1 - 8.87T + 79T^{2} \) |
| 83 | \( 1 + 8.08T + 83T^{2} \) |
| 89 | \( 1 + 3.46T + 89T^{2} \) |
| 97 | \( 1 - 3.92T + 97T^{2} \) |
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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
−7.50785643546130943718561892924, −7.20559172483875127733284640416, −6.46393662083771393661885691692, −5.68880319642136337469689129010, −5.12509229246776801920142458086, −4.02361101956437439050934480988, −3.43409918329740647999453363940, −2.62760387596380505904434562389, −2.41834751852809128365460118943, −0.827655450974849209327545471185,
0.827655450974849209327545471185, 2.41834751852809128365460118943, 2.62760387596380505904434562389, 3.43409918329740647999453363940, 4.02361101956437439050934480988, 5.12509229246776801920142458086, 5.68880319642136337469689129010, 6.46393662083771393661885691692, 7.20559172483875127733284640416, 7.50785643546130943718561892924
