| L(s) = 1 | − 2-s + 4-s + 5-s − 7-s − 8-s − 3·9-s − 10-s − 2·11-s − 4·13-s + 14-s + 16-s − 6·17-s + 3·18-s + 4·19-s + 20-s + 2·22-s + 25-s + 4·26-s − 28-s + 6·29-s − 10·31-s − 32-s + 6·34-s − 35-s − 3·36-s + 6·37-s − 4·38-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.377·7-s − 0.353·8-s − 9-s − 0.316·10-s − 0.603·11-s − 1.10·13-s + 0.267·14-s + 1/4·16-s − 1.45·17-s + 0.707·18-s + 0.917·19-s + 0.223·20-s + 0.426·22-s + 1/5·25-s + 0.784·26-s − 0.188·28-s + 1.11·29-s − 1.79·31-s − 0.176·32-s + 1.02·34-s − 0.169·35-s − 1/2·36-s + 0.986·37-s − 0.648·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−15.08426104883545, −14.78119037851476, −14.00269201935652, −13.70389754675789, −12.99609980072342, −12.46363218913066, −11.99098869838197, −11.26684879634130, −10.89483102528226, −10.40609546645463, −9.656604344484223, −9.285257518924554, −8.929557762417564, −8.123759888010009, −7.693445432793387, −7.036679316921902, −6.507785573042022, −5.876039327506047, −5.285404146990304, −4.767216258419301, −3.825396734028919, −2.925687506749134, −2.525676361066997, −1.963273253231971, −0.7640719306039866, 0,
0.7640719306039866, 1.963273253231971, 2.525676361066997, 2.925687506749134, 3.825396734028919, 4.767216258419301, 5.285404146990304, 5.876039327506047, 6.507785573042022, 7.036679316921902, 7.693445432793387, 8.123759888010009, 8.929557762417564, 9.285257518924554, 9.656604344484223, 10.40609546645463, 10.89483102528226, 11.26684879634130, 11.99098869838197, 12.46363218913066, 12.99609980072342, 13.70389754675789, 14.00269201935652, 14.78119037851476, 15.08426104883545
