L(s) = 1 | + 2.13·2-s + 2.03·3-s + 2.55·4-s + 5-s + 4.34·6-s + 2.85·7-s + 1.18·8-s + 1.13·9-s + 2.13·10-s − 11-s + 5.19·12-s − 2.04·13-s + 6.10·14-s + 2.03·15-s − 2.57·16-s + 7.51·17-s + 2.41·18-s + 1.92·19-s + 2.55·20-s + 5.81·21-s − 2.13·22-s + 2.62·23-s + 2.41·24-s + 25-s − 4.36·26-s − 3.79·27-s + 7.31·28-s + ⋯ |
L(s) = 1 | + 1.50·2-s + 1.17·3-s + 1.27·4-s + 0.447·5-s + 1.77·6-s + 1.08·7-s + 0.420·8-s + 0.377·9-s + 0.675·10-s − 0.301·11-s + 1.50·12-s − 0.566·13-s + 1.63·14-s + 0.524·15-s − 0.643·16-s + 1.82·17-s + 0.570·18-s + 0.441·19-s + 0.571·20-s + 1.26·21-s − 0.455·22-s + 0.547·23-s + 0.493·24-s + 0.200·25-s − 0.855·26-s − 0.730·27-s + 1.38·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(8.012331750\) |
\(L(\frac12)\) |
\(\approx\) |
\(8.012331750\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 73 | \( 1 + T \) |
good | 2 | \( 1 - 2.13T + 2T^{2} \) |
| 3 | \( 1 - 2.03T + 3T^{2} \) |
| 7 | \( 1 - 2.85T + 7T^{2} \) |
| 13 | \( 1 + 2.04T + 13T^{2} \) |
| 17 | \( 1 - 7.51T + 17T^{2} \) |
| 19 | \( 1 - 1.92T + 19T^{2} \) |
| 23 | \( 1 - 2.62T + 23T^{2} \) |
| 29 | \( 1 + 1.22T + 29T^{2} \) |
| 31 | \( 1 - 5.61T + 31T^{2} \) |
| 37 | \( 1 + 5.83T + 37T^{2} \) |
| 41 | \( 1 + 11.6T + 41T^{2} \) |
| 43 | \( 1 - 7.76T + 43T^{2} \) |
| 47 | \( 1 - 12.1T + 47T^{2} \) |
| 53 | \( 1 + 0.268T + 53T^{2} \) |
| 59 | \( 1 - 9.95T + 59T^{2} \) |
| 61 | \( 1 + 1.70T + 61T^{2} \) |
| 67 | \( 1 + 13.4T + 67T^{2} \) |
| 71 | \( 1 + 8.37T + 71T^{2} \) |
| 79 | \( 1 - 11.4T + 79T^{2} \) |
| 83 | \( 1 + 3.93T + 83T^{2} \) |
| 89 | \( 1 + 5.55T + 89T^{2} \) |
| 97 | \( 1 + 15.8T + 97T^{2} \) |
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
−8.360534477998771548364312738541, −7.60626824786557623513196510031, −7.06285215058616588243812913604, −5.82410894401491489587806811877, −5.35224257234357206739042839093, −4.70770221531974360217046640113, −3.76612214581121271222509294697, −3.03445729834286226098706007593, −2.43929041892932388334565780290, −1.43850481014294529635167564190,
1.43850481014294529635167564190, 2.43929041892932388334565780290, 3.03445729834286226098706007593, 3.76612214581121271222509294697, 4.70770221531974360217046640113, 5.35224257234357206739042839093, 5.82410894401491489587806811877, 7.06285215058616588243812913604, 7.60626824786557623513196510031, 8.360534477998771548364312738541