L(s) = 1 | + 2-s + 2.08·3-s + 4-s − 0.817·5-s + 2.08·6-s − 0.777·7-s + 8-s + 1.35·9-s − 0.817·10-s + 0.583·11-s + 2.08·12-s + 6.54·13-s − 0.777·14-s − 1.70·15-s + 16-s − 2.73·17-s + 1.35·18-s − 19-s − 0.817·20-s − 1.62·21-s + 0.583·22-s + 6.31·23-s + 2.08·24-s − 4.33·25-s + 6.54·26-s − 3.42·27-s − 0.777·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.20·3-s + 0.5·4-s − 0.365·5-s + 0.852·6-s − 0.293·7-s + 0.353·8-s + 0.453·9-s − 0.258·10-s + 0.175·11-s + 0.602·12-s + 1.81·13-s − 0.207·14-s − 0.440·15-s + 0.250·16-s − 0.664·17-s + 0.320·18-s − 0.229·19-s − 0.182·20-s − 0.354·21-s + 0.124·22-s + 1.31·23-s + 0.426·24-s − 0.866·25-s + 1.28·26-s − 0.659·27-s − 0.146·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.223446095\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.223446095\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| 211 | \( 1 + T \) |
good | 3 | \( 1 - 2.08T + 3T^{2} \) |
| 5 | \( 1 + 0.817T + 5T^{2} \) |
| 7 | \( 1 + 0.777T + 7T^{2} \) |
| 11 | \( 1 - 0.583T + 11T^{2} \) |
| 13 | \( 1 - 6.54T + 13T^{2} \) |
| 17 | \( 1 + 2.73T + 17T^{2} \) |
| 23 | \( 1 - 6.31T + 23T^{2} \) |
| 29 | \( 1 - 5.06T + 29T^{2} \) |
| 31 | \( 1 - 8.73T + 31T^{2} \) |
| 37 | \( 1 - 0.521T + 37T^{2} \) |
| 41 | \( 1 - 7.07T + 41T^{2} \) |
| 43 | \( 1 - 3.17T + 43T^{2} \) |
| 47 | \( 1 + 7.30T + 47T^{2} \) |
| 53 | \( 1 - 2.58T + 53T^{2} \) |
| 59 | \( 1 + 6.28T + 59T^{2} \) |
| 61 | \( 1 - 3.79T + 61T^{2} \) |
| 67 | \( 1 + 12.4T + 67T^{2} \) |
| 71 | \( 1 - 14.2T + 71T^{2} \) |
| 73 | \( 1 + 5.81T + 73T^{2} \) |
| 79 | \( 1 + 5.18T + 79T^{2} \) |
| 83 | \( 1 - 16.9T + 83T^{2} \) |
| 89 | \( 1 - 4.08T + 89T^{2} \) |
| 97 | \( 1 + 11.4T + 97T^{2} \) |
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
−8.005767128418230107118382164530, −7.11354831996893233422912223235, −6.35418253043370523396925347814, −5.94815051027155967533187188430, −4.73784564568850633235801950628, −4.16318074140235745737141681369, −3.39105869280488386644637797937, −2.97633562481856728848608596996, −2.05420241396626604425453505399, −0.994585898524853195793548759160,
0.994585898524853195793548759160, 2.05420241396626604425453505399, 2.97633562481856728848608596996, 3.39105869280488386644637797937, 4.16318074140235745737141681369, 4.73784564568850633235801950628, 5.94815051027155967533187188430, 6.35418253043370523396925347814, 7.11354831996893233422912223235, 8.005767128418230107118382164530