| L(s) = 1 | + 2-s + 3-s + 4-s + 1.23·5-s + 6-s − 4.47·7-s + 8-s + 9-s + 1.23·10-s − 5.23·11-s + 12-s + 4.47·13-s − 4.47·14-s + 1.23·15-s + 16-s − 4·17-s + 18-s + 5.70·19-s + 1.23·20-s − 4.47·21-s − 5.23·22-s + 23-s + 24-s − 3.47·25-s + 4.47·26-s + 27-s − 4.47·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.552·5-s + 0.408·6-s − 1.69·7-s + 0.353·8-s + 0.333·9-s + 0.390·10-s − 1.57·11-s + 0.288·12-s + 1.24·13-s − 1.19·14-s + 0.319·15-s + 0.250·16-s − 0.970·17-s + 0.235·18-s + 1.30·19-s + 0.276·20-s − 0.975·21-s − 1.11·22-s + 0.208·23-s + 0.204·24-s − 0.694·25-s + 0.877·26-s + 0.192·27-s − 0.845·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.729189778\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.729189778\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 - 1.23T + 5T^{2} \) |
| 7 | \( 1 + 4.47T + 7T^{2} \) |
| 11 | \( 1 + 5.23T + 11T^{2} \) |
| 13 | \( 1 - 4.47T + 13T^{2} \) |
| 17 | \( 1 + 4T + 17T^{2} \) |
| 19 | \( 1 - 5.70T + 19T^{2} \) |
| 29 | \( 1 + 4.47T + 29T^{2} \) |
| 31 | \( 1 + 2.47T + 31T^{2} \) |
| 37 | \( 1 - 11.2T + 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 4.76T + 43T^{2} \) |
| 47 | \( 1 - 4T + 47T^{2} \) |
| 53 | \( 1 - 5.23T + 53T^{2} \) |
| 59 | \( 1 + 8.94T + 59T^{2} \) |
| 61 | \( 1 - 0.763T + 61T^{2} \) |
| 67 | \( 1 - 9.70T + 67T^{2} \) |
| 71 | \( 1 - 8.94T + 71T^{2} \) |
| 73 | \( 1 + 4.47T + 73T^{2} \) |
| 79 | \( 1 - 4.47T + 79T^{2} \) |
| 83 | \( 1 - 13.2T + 83T^{2} \) |
| 89 | \( 1 + 10.4T + 89T^{2} \) |
| 97 | \( 1 - 0.472T + 97T^{2} \) |
| 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
−13.40296048666462235060175432377, −12.70448990159139527853431844986, −11.14362763232515860468997403072, −10.05246709441275187518308746240, −9.202728600148604334408875181294, −7.72541587566694873137775243372, −6.47449226925070426969624600597, −5.49721333017287637529615359721, −3.67905541281235520547812236276, −2.59871371115670108510936022230,
2.59871371115670108510936022230, 3.67905541281235520547812236276, 5.49721333017287637529615359721, 6.47449226925070426969624600597, 7.72541587566694873137775243372, 9.202728600148604334408875181294, 10.05246709441275187518308746240, 11.14362763232515860468997403072, 12.70448990159139527853431844986, 13.40296048666462235060175432377
