| L(s) = 1 | − 4.59·2-s − 3·3-s + 13.1·4-s + 5·5-s + 13.7·6-s − 20.6·7-s − 23.4·8-s + 9·9-s − 22.9·10-s − 39.3·12-s + 15.6·13-s + 94.8·14-s − 15·15-s + 3.04·16-s − 72.9·17-s − 41.3·18-s − 61.0·19-s + 65.5·20-s + 61.9·21-s − 13.6·23-s + 70.4·24-s + 25·25-s − 71.9·26-s − 27·27-s − 270.·28-s + 31.4·29-s + 68.9·30-s + ⋯ |
| L(s) = 1 | − 1.62·2-s − 0.577·3-s + 1.63·4-s + 0.447·5-s + 0.937·6-s − 1.11·7-s − 1.03·8-s + 0.333·9-s − 0.726·10-s − 0.946·12-s + 0.334·13-s + 1.81·14-s − 0.258·15-s + 0.0475·16-s − 1.04·17-s − 0.541·18-s − 0.737·19-s + 0.733·20-s + 0.643·21-s − 0.123·23-s + 0.599·24-s + 0.200·25-s − 0.542·26-s − 0.192·27-s − 1.82·28-s + 0.201·29-s + 0.419·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.2608577394\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2608577394\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 5 | \( 1 - 5T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 4.59T + 8T^{2} \) |
| 7 | \( 1 + 20.6T + 343T^{2} \) |
| 13 | \( 1 - 15.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 72.9T + 4.91e3T^{2} \) |
| 19 | \( 1 + 61.0T + 6.85e3T^{2} \) |
| 23 | \( 1 + 13.6T + 1.21e4T^{2} \) |
| 29 | \( 1 - 31.4T + 2.43e4T^{2} \) |
| 31 | \( 1 + 243.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 65.4T + 5.06e4T^{2} \) |
| 41 | \( 1 - 109.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 121.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 519.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 542.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 109.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 89.6T + 2.26e5T^{2} \) |
| 67 | \( 1 - 488.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 837.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 351.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 831.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.38e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.52e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 426.T + 9.12e5T^{2} \) |
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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
−9.082254537570738414961006770985, −8.326621936422461816483293605958, −7.36220383708884104814425660374, −6.50159655936816067446821691173, −6.27652538351514078849404100547, −4.99713438011047284156720157922, −3.75258130211244852141142813809, −2.47523608031866981281663660257, −1.55694626157780147573746113520, −0.31735718401735330732760290016,
0.31735718401735330732760290016, 1.55694626157780147573746113520, 2.47523608031866981281663660257, 3.75258130211244852141142813809, 4.99713438011047284156720157922, 6.27652538351514078849404100547, 6.50159655936816067446821691173, 7.36220383708884104814425660374, 8.326621936422461816483293605958, 9.082254537570738414961006770985
