| L(s) = 1 | + 2-s − 1.44·3-s + 4-s − 0.880·5-s − 1.44·6-s − 5.04·7-s + 8-s − 0.915·9-s − 0.880·10-s + 11-s − 1.44·12-s + 1.58·13-s − 5.04·14-s + 1.27·15-s + 16-s − 3.33·17-s − 0.915·18-s − 2.18·19-s − 0.880·20-s + 7.28·21-s + 22-s + 0.594·23-s − 1.44·24-s − 4.22·25-s + 1.58·26-s + 5.65·27-s − 5.04·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.833·3-s + 0.5·4-s − 0.393·5-s − 0.589·6-s − 1.90·7-s + 0.353·8-s − 0.305·9-s − 0.278·10-s + 0.301·11-s − 0.416·12-s + 0.440·13-s − 1.34·14-s + 0.328·15-s + 0.250·16-s − 0.810·17-s − 0.215·18-s − 0.501·19-s − 0.196·20-s + 1.59·21-s + 0.213·22-s + 0.124·23-s − 0.294·24-s − 0.845·25-s + 0.311·26-s + 1.08·27-s − 0.954·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8629201454\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8629201454\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 197 | \( 1 - T \) |
| good | 3 | \( 1 + 1.44T + 3T^{2} \) |
| 5 | \( 1 + 0.880T + 5T^{2} \) |
| 7 | \( 1 + 5.04T + 7T^{2} \) |
| 13 | \( 1 - 1.58T + 13T^{2} \) |
| 17 | \( 1 + 3.33T + 17T^{2} \) |
| 19 | \( 1 + 2.18T + 19T^{2} \) |
| 23 | \( 1 - 0.594T + 23T^{2} \) |
| 29 | \( 1 - 6.81T + 29T^{2} \) |
| 31 | \( 1 + 8.51T + 31T^{2} \) |
| 37 | \( 1 + 11.5T + 37T^{2} \) |
| 41 | \( 1 + 0.0999T + 41T^{2} \) |
| 43 | \( 1 + 7.64T + 43T^{2} \) |
| 47 | \( 1 - 10.6T + 47T^{2} \) |
| 53 | \( 1 + 8.79T + 53T^{2} \) |
| 59 | \( 1 - 12.2T + 59T^{2} \) |
| 61 | \( 1 + 6.41T + 61T^{2} \) |
| 67 | \( 1 - 2.66T + 67T^{2} \) |
| 71 | \( 1 - 8.78T + 71T^{2} \) |
| 73 | \( 1 - 0.0781T + 73T^{2} \) |
| 79 | \( 1 + 0.149T + 79T^{2} \) |
| 83 | \( 1 - 6.73T + 83T^{2} \) |
| 89 | \( 1 + 13.1T + 89T^{2} \) |
| 97 | \( 1 - 7.38T + 97T^{2} \) |
| 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
−8.490937575702757844031728915986, −7.19151284387035615196134882550, −6.70913798969150844883972351355, −6.15538474690380758396115369875, −5.58858385394920368875899857528, −4.64883060746034982214543987461, −3.69097535225820005456164014668, −3.26691374983002844539404457499, −2.13964848240007130262332233910, −0.46011018605260304745211018784,
0.46011018605260304745211018784, 2.13964848240007130262332233910, 3.26691374983002844539404457499, 3.69097535225820005456164014668, 4.64883060746034982214543987461, 5.58858385394920368875899857528, 6.15538474690380758396115369875, 6.70913798969150844883972351355, 7.19151284387035615196134882550, 8.490937575702757844031728915986
