L(s) = 1 | + 0.473·2-s + 2.13·3-s − 1.77·4-s + 5-s + 1.01·6-s + 3.05·7-s − 1.78·8-s + 1.57·9-s + 0.473·10-s − 1.37·11-s − 3.79·12-s − 4.61·13-s + 1.44·14-s + 2.13·15-s + 2.70·16-s − 17-s + 0.745·18-s − 4.02·19-s − 1.77·20-s + 6.53·21-s − 0.649·22-s + 0.0167·23-s − 3.82·24-s + 25-s − 2.18·26-s − 3.04·27-s − 5.42·28-s + ⋯ |
L(s) = 1 | + 0.334·2-s + 1.23·3-s − 0.887·4-s + 0.447·5-s + 0.413·6-s + 1.15·7-s − 0.632·8-s + 0.524·9-s + 0.149·10-s − 0.413·11-s − 1.09·12-s − 1.28·13-s + 0.386·14-s + 0.552·15-s + 0.676·16-s − 0.242·17-s + 0.175·18-s − 0.924·19-s − 0.397·20-s + 1.42·21-s − 0.138·22-s + 0.00349·23-s − 0.780·24-s + 0.200·25-s − 0.428·26-s − 0.586·27-s − 1.02·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6035 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 71 | \( 1 - T \) |
good | 2 | \( 1 - 0.473T + 2T^{2} \) |
| 3 | \( 1 - 2.13T + 3T^{2} \) |
| 7 | \( 1 - 3.05T + 7T^{2} \) |
| 11 | \( 1 + 1.37T + 11T^{2} \) |
| 13 | \( 1 + 4.61T + 13T^{2} \) |
| 19 | \( 1 + 4.02T + 19T^{2} \) |
| 23 | \( 1 - 0.0167T + 23T^{2} \) |
| 29 | \( 1 + 6.76T + 29T^{2} \) |
| 31 | \( 1 - 3.88T + 31T^{2} \) |
| 37 | \( 1 + 11.0T + 37T^{2} \) |
| 41 | \( 1 + 4.10T + 41T^{2} \) |
| 43 | \( 1 - 2.43T + 43T^{2} \) |
| 47 | \( 1 + 4.17T + 47T^{2} \) |
| 53 | \( 1 - 5.22T + 53T^{2} \) |
| 59 | \( 1 + 4.68T + 59T^{2} \) |
| 61 | \( 1 + 2.87T + 61T^{2} \) |
| 67 | \( 1 + 11.0T + 67T^{2} \) |
| 73 | \( 1 - 7.48T + 73T^{2} \) |
| 79 | \( 1 - 6.98T + 79T^{2} \) |
| 83 | \( 1 + 7.21T + 83T^{2} \) |
| 89 | \( 1 + 4.20T + 89T^{2} \) |
| 97 | \( 1 - 1.53T + 97T^{2} \) |
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
−7.80942120799227264389975339213, −7.34026787022467119666544101144, −6.19718663302108535562674794356, −5.23313017663441653785266393989, −4.85977663100618083839224723926, −4.06438886980546510155890303753, −3.22197933720080577764991866635, −2.35009542562794982123421552895, −1.71140875723211620530112169898, 0,
1.71140875723211620530112169898, 2.35009542562794982123421552895, 3.22197933720080577764991866635, 4.06438886980546510155890303753, 4.85977663100618083839224723926, 5.23313017663441653785266393989, 6.19718663302108535562674794356, 7.34026787022467119666544101144, 7.80942120799227264389975339213