| L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s + 7-s − 8-s + 9-s + 10-s − 11-s + 12-s + 13-s − 14-s − 15-s + 16-s − 18-s − 4·19-s − 20-s + 21-s + 22-s + 6·23-s − 24-s + 25-s − 26-s + 27-s + 28-s + 6·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.301·11-s + 0.288·12-s + 0.277·13-s − 0.267·14-s − 0.258·15-s + 1/4·16-s − 0.235·18-s − 0.917·19-s − 0.223·20-s + 0.218·21-s + 0.213·22-s + 1.25·23-s − 0.204·24-s + 1/5·25-s − 0.196·26-s + 0.192·27-s + 0.188·28-s + 1.11·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.902286403\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.902286403\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−15.10760656154231, −14.76217479281877, −14.24154824660104, −13.47213717574050, −13.07155473785902, −12.41782239157840, −11.92290510506345, −11.17674086669022, −10.89511121723589, −10.22878740709051, −9.752092223372899, −8.976949504520921, −8.578608021952614, −8.162966850856156, −7.613232770314351, −6.939888823543095, −6.513326515688043, −5.709518204571794, −4.833799269907325, −4.404581893165125, −3.501264478920813, −2.909562520161407, −2.238048122387463, −1.399751442668883, −0.5998719222902012,
0.5998719222902012, 1.399751442668883, 2.238048122387463, 2.909562520161407, 3.501264478920813, 4.404581893165125, 4.833799269907325, 5.709518204571794, 6.513326515688043, 6.939888823543095, 7.613232770314351, 8.162966850856156, 8.578608021952614, 8.976949504520921, 9.752092223372899, 10.22878740709051, 10.89511121723589, 11.17674086669022, 11.92290510506345, 12.41782239157840, 13.07155473785902, 13.47213717574050, 14.24154824660104, 14.76217479281877, 15.10760656154231
