| L(s) = 1 | + 0.0419·2-s − 3.02·3-s − 1.99·4-s − 1.74·5-s − 0.126·6-s + 2.05·7-s − 0.167·8-s + 6.16·9-s − 0.0730·10-s + 1.85·11-s + 6.05·12-s + 2.57·13-s + 0.0861·14-s + 5.27·15-s + 3.98·16-s − 5.23·17-s + 0.258·18-s − 0.379·19-s + 3.48·20-s − 6.21·21-s + 0.0776·22-s + 3.63·23-s + 0.507·24-s − 1.96·25-s + 0.108·26-s − 9.58·27-s − 4.10·28-s + ⋯ |
| L(s) = 1 | + 0.0296·2-s − 1.74·3-s − 0.999·4-s − 0.779·5-s − 0.0518·6-s + 0.776·7-s − 0.0592·8-s + 2.05·9-s − 0.0231·10-s + 0.558·11-s + 1.74·12-s + 0.714·13-s + 0.0230·14-s + 1.36·15-s + 0.997·16-s − 1.26·17-s + 0.0609·18-s − 0.0869·19-s + 0.778·20-s − 1.35·21-s + 0.0165·22-s + 0.757·23-s + 0.103·24-s − 0.393·25-s + 0.0212·26-s − 1.84·27-s − 0.775·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 29 | \( 1 \) |
| good | 2 | \( 1 - 0.0419T + 2T^{2} \) |
| 3 | \( 1 + 3.02T + 3T^{2} \) |
| 5 | \( 1 + 1.74T + 5T^{2} \) |
| 7 | \( 1 - 2.05T + 7T^{2} \) |
| 11 | \( 1 - 1.85T + 11T^{2} \) |
| 13 | \( 1 - 2.57T + 13T^{2} \) |
| 17 | \( 1 + 5.23T + 17T^{2} \) |
| 19 | \( 1 + 0.379T + 19T^{2} \) |
| 23 | \( 1 - 3.63T + 23T^{2} \) |
| 31 | \( 1 + 2.36T + 31T^{2} \) |
| 37 | \( 1 - 6.56T + 37T^{2} \) |
| 41 | \( 1 + 9.44T + 41T^{2} \) |
| 43 | \( 1 - 0.995T + 43T^{2} \) |
| 47 | \( 1 + 10.1T + 47T^{2} \) |
| 53 | \( 1 - 9.93T + 53T^{2} \) |
| 59 | \( 1 + 6.24T + 59T^{2} \) |
| 61 | \( 1 - 7.98T + 61T^{2} \) |
| 67 | \( 1 + 4.13T + 67T^{2} \) |
| 71 | \( 1 + 2.28T + 71T^{2} \) |
| 73 | \( 1 - 9.06T + 73T^{2} \) |
| 79 | \( 1 + 10.2T + 79T^{2} \) |
| 83 | \( 1 + 15.6T + 83T^{2} \) |
| 89 | \( 1 - 9.61T + 89T^{2} \) |
| 97 | \( 1 + 5.17T + 97T^{2} \) |
| 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
−9.949420717437631188250207082872, −8.906250039555293352026517729327, −8.112234510432481801575492425329, −7.02363435936436704330980807109, −6.16702827181305489230412205532, −5.18444511262223159688928596805, −4.52263523143849229955120190750, −3.81958927161261144874096920569, −1.32067983669095626157272169336, 0,
1.32067983669095626157272169336, 3.81958927161261144874096920569, 4.52263523143849229955120190750, 5.18444511262223159688928596805, 6.16702827181305489230412205532, 7.02363435936436704330980807109, 8.112234510432481801575492425329, 8.906250039555293352026517729327, 9.949420717437631188250207082872
