| L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s − 1.60·7-s + 8-s + 9-s + 10-s − 3.49·11-s + 12-s − 2.98·13-s − 1.60·14-s + 15-s + 16-s + 18-s + 2.37·19-s + 20-s − 1.60·21-s − 3.49·22-s − 0.784·23-s + 24-s + 25-s − 2.98·26-s + 27-s − 1.60·28-s + 10.1·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.447·5-s + 0.408·6-s − 0.605·7-s + 0.353·8-s + 0.333·9-s + 0.316·10-s − 1.05·11-s + 0.288·12-s − 0.828·13-s − 0.428·14-s + 0.258·15-s + 0.250·16-s + 0.235·18-s + 0.544·19-s + 0.223·20-s − 0.349·21-s − 0.744·22-s − 0.163·23-s + 0.204·24-s + 0.200·25-s − 0.585·26-s + 0.192·27-s − 0.302·28-s + 1.87·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.015848213\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.015848213\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 7 | \( 1 + 1.60T + 7T^{2} \) |
| 11 | \( 1 + 3.49T + 11T^{2} \) |
| 13 | \( 1 + 2.98T + 13T^{2} \) |
| 19 | \( 1 - 2.37T + 19T^{2} \) |
| 23 | \( 1 + 0.784T + 23T^{2} \) |
| 29 | \( 1 - 10.1T + 29T^{2} \) |
| 31 | \( 1 - 6.28T + 31T^{2} \) |
| 37 | \( 1 - 11.6T + 37T^{2} \) |
| 41 | \( 1 - 4.85T + 41T^{2} \) |
| 43 | \( 1 + 5.92T + 43T^{2} \) |
| 47 | \( 1 - 1.44T + 47T^{2} \) |
| 53 | \( 1 + 6.25T + 53T^{2} \) |
| 59 | \( 1 - 3.25T + 59T^{2} \) |
| 61 | \( 1 + 6.21T + 61T^{2} \) |
| 67 | \( 1 - 7.98T + 67T^{2} \) |
| 71 | \( 1 - 2.35T + 71T^{2} \) |
| 73 | \( 1 - 3.79T + 73T^{2} \) |
| 79 | \( 1 + 4.37T + 79T^{2} \) |
| 83 | \( 1 - 12.0T + 83T^{2} \) |
| 89 | \( 1 + 2.14T + 89T^{2} \) |
| 97 | \( 1 + 0.268T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.906898380726289877887934423418, −6.93219797764404487800603663723, −6.40438495299473275982799529134, −5.67642929662504712572613405862, −4.83181703906267114194962178931, −4.42701916576740484874137700708, −3.20655975775778356013224854906, −2.80300614647273740041066198385, −2.15364567811618317093378510554, −0.836410230288148795335547286237,
0.836410230288148795335547286237, 2.15364567811618317093378510554, 2.80300614647273740041066198385, 3.20655975775778356013224854906, 4.42701916576740484874137700708, 4.83181703906267114194962178931, 5.67642929662504712572613405862, 6.40438495299473275982799529134, 6.93219797764404487800603663723, 7.906898380726289877887934423418
