L(s) = 1 | − 2-s + 1.55·3-s + 4-s + 2.24·5-s − 1.55·6-s + 2.27·7-s − 8-s − 0.579·9-s − 2.24·10-s + 1.55·12-s − 2.75·13-s − 2.27·14-s + 3.48·15-s + 16-s + 0.571·17-s + 0.579·18-s + 19-s + 2.24·20-s + 3.54·21-s + 9.47·23-s − 1.55·24-s + 0.0190·25-s + 2.75·26-s − 5.56·27-s + 2.27·28-s − 1.99·29-s − 3.48·30-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.898·3-s + 0.5·4-s + 1.00·5-s − 0.635·6-s + 0.860·7-s − 0.353·8-s − 0.193·9-s − 0.708·10-s + 0.449·12-s − 0.764·13-s − 0.608·14-s + 0.900·15-s + 0.250·16-s + 0.138·17-s + 0.136·18-s + 0.229·19-s + 0.500·20-s + 0.773·21-s + 1.97·23-s − 0.317·24-s + 0.00381·25-s + 0.540·26-s − 1.07·27-s + 0.430·28-s − 0.370·29-s − 0.636·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.600107316\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.600107316\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 3 | \( 1 - 1.55T + 3T^{2} \) |
| 5 | \( 1 - 2.24T + 5T^{2} \) |
| 7 | \( 1 - 2.27T + 7T^{2} \) |
| 13 | \( 1 + 2.75T + 13T^{2} \) |
| 17 | \( 1 - 0.571T + 17T^{2} \) |
| 23 | \( 1 - 9.47T + 23T^{2} \) |
| 29 | \( 1 + 1.99T + 29T^{2} \) |
| 31 | \( 1 - 8.92T + 31T^{2} \) |
| 37 | \( 1 - 7.48T + 37T^{2} \) |
| 41 | \( 1 - 0.0862T + 41T^{2} \) |
| 43 | \( 1 + 5.53T + 43T^{2} \) |
| 47 | \( 1 + 9.68T + 47T^{2} \) |
| 53 | \( 1 - 4.40T + 53T^{2} \) |
| 59 | \( 1 - 8.02T + 59T^{2} \) |
| 61 | \( 1 + 3.73T + 61T^{2} \) |
| 67 | \( 1 - 6.91T + 67T^{2} \) |
| 71 | \( 1 - 4.91T + 71T^{2} \) |
| 73 | \( 1 - 10.7T + 73T^{2} \) |
| 79 | \( 1 - 5.92T + 79T^{2} \) |
| 83 | \( 1 - 16.8T + 83T^{2} \) |
| 89 | \( 1 - 5.14T + 89T^{2} \) |
| 97 | \( 1 - 3.98T + 97T^{2} \) |
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
−8.204124049922839523471929527894, −7.967675719890254929834291987874, −7.01484469873582320401432101401, −6.31027561008692509507535127713, −5.31613341483422660927603230904, −4.79716361638135898305677494654, −3.42940436607693590333116717236, −2.60290884540422844389949530232, −2.02847611235905995882854130090, −0.989866747133183708458559075794,
0.989866747133183708458559075794, 2.02847611235905995882854130090, 2.60290884540422844389949530232, 3.42940436607693590333116717236, 4.79716361638135898305677494654, 5.31613341483422660927603230904, 6.31027561008692509507535127713, 7.01484469873582320401432101401, 7.967675719890254929834291987874, 8.204124049922839523471929527894