| L(s) = 1 | + 2-s − 3-s + 4-s − 4·5-s − 6-s − 7-s + 8-s + 9-s − 4·10-s − 12-s − 2·13-s − 14-s + 4·15-s + 16-s − 2·17-s + 18-s + 2·19-s − 4·20-s + 21-s + 23-s − 24-s + 11·25-s − 2·26-s − 27-s − 28-s + 6·29-s + 4·30-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 + 1/3·9-s − 1.26·10-s − 0.288·12-s − 0.554·13-s − 0.267·14-s + 1.03·15-s + 1/4·16-s − 0.485·17-s + 0.235·18-s + 0.458·19-s − 0.894·20-s + 0.218·21-s + 0.208·23-s − 0.204·24-s + 11/5·25-s − 0.392·26-s − 0.192·27-s − 0.188·28-s + 1.11·29-s + 0.730·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 116886 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116886 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.250851850\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.250851850\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−13.47059560612836, −12.92500398630310, −12.51523238210318, −12.18953539546984, −11.73477718649594, −11.25886560083751, −10.91672945322062, −10.46903994900100, −9.680108543390050, −9.279140373091264, −8.481125864593586, −8.022786436930642, −7.455936361934135, −7.190456972981976, −6.523644876705074, −6.146663674658675, −5.355410454592018, −4.796180975962054, −4.412116122337128, −3.941978128676808, −3.329977685372178, −2.802443098905422, −2.127896168464543, −0.8879825040069972, −0.5626999360451246,
0.5626999360451246, 0.8879825040069972, 2.127896168464543, 2.802443098905422, 3.329977685372178, 3.941978128676808, 4.412116122337128, 4.796180975962054, 5.355410454592018, 6.146663674658675, 6.523644876705074, 7.190456972981976, 7.455936361934135, 8.022786436930642, 8.481125864593586, 9.279140373091264, 9.680108543390050, 10.46903994900100, 10.91672945322062, 11.25886560083751, 11.73477718649594, 12.18953539546984, 12.51523238210318, 12.92500398630310, 13.47059560612836
