| L(s) = 1 | + 2-s − 0.352·3-s + 4-s − 2.30·5-s − 0.352·6-s + 2.78·7-s + 8-s − 2.87·9-s − 2.30·10-s − 11-s − 0.352·12-s + 2.55·13-s + 2.78·14-s + 0.811·15-s + 16-s − 2.24·17-s − 2.87·18-s + 1.37·19-s − 2.30·20-s − 0.981·21-s − 22-s + 4.45·23-s − 0.352·24-s + 0.296·25-s + 2.55·26-s + 2.07·27-s + 2.78·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.203·3-s + 0.5·4-s − 1.02·5-s − 0.143·6-s + 1.05·7-s + 0.353·8-s − 0.958·9-s − 0.727·10-s − 0.301·11-s − 0.101·12-s + 0.707·13-s + 0.743·14-s + 0.209·15-s + 0.250·16-s − 0.544·17-s − 0.677·18-s + 0.314·19-s − 0.514·20-s − 0.214·21-s − 0.213·22-s + 0.928·23-s − 0.0719·24-s + 0.0593·25-s + 0.500·26-s + 0.398·27-s + 0.525·28-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
| \(L(1)\) |
\(\approx\) |
\(2.385730210\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.385730210\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.352T + 3T^{2} \) |
| 5 | \( 1 + 2.30T + 5T^{2} \) |
| 7 | \( 1 - 2.78T + 7T^{2} \) |
| 13 | \( 1 - 2.55T + 13T^{2} \) |
| 17 | \( 1 + 2.24T + 17T^{2} \) |
| 19 | \( 1 - 1.37T + 19T^{2} \) |
| 23 | \( 1 - 4.45T + 23T^{2} \) |
| 29 | \( 1 - 1.67T + 29T^{2} \) |
| 31 | \( 1 - 4.85T + 31T^{2} \) |
| 37 | \( 1 - 0.216T + 37T^{2} \) |
| 41 | \( 1 - 5.79T + 41T^{2} \) |
| 43 | \( 1 + 6.26T + 43T^{2} \) |
| 47 | \( 1 + 11.3T + 47T^{2} \) |
| 53 | \( 1 + 4.29T + 53T^{2} \) |
| 59 | \( 1 + 2.83T + 59T^{2} \) |
| 61 | \( 1 - 11.9T + 61T^{2} \) |
| 67 | \( 1 - 11.4T + 67T^{2} \) |
| 71 | \( 1 - 7.34T + 71T^{2} \) |
| 73 | \( 1 - 9.18T + 73T^{2} \) |
| 79 | \( 1 - 7.70T + 79T^{2} \) |
| 83 | \( 1 - 8.24T + 83T^{2} \) |
| 89 | \( 1 - 4.26T + 89T^{2} \) |
| 97 | \( 1 - 9.36T + 97T^{2} \) |
| 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
−8.162073171751248991754667908028, −7.82246772846523098711033096670, −6.78161971171128285677159591544, −6.16945855610255738673293230329, −5.07641242435386987432836101836, −4.85530739570857038126460270224, −3.81160723524296150965362979447, −3.13922426674413548591583996589, −2.10851077864328387835161942957, −0.791716400065104803343192292178,
0.791716400065104803343192292178, 2.10851077864328387835161942957, 3.13922426674413548591583996589, 3.81160723524296150965362979447, 4.85530739570857038126460270224, 5.07641242435386987432836101836, 6.16945855610255738673293230329, 6.78161971171128285677159591544, 7.82246772846523098711033096670, 8.162073171751248991754667908028
