| L(s) = 1 | − 2.43·2-s + 3.36·3-s + 3.93·4-s + 0.698·5-s − 8.19·6-s + 2.30·7-s − 4.70·8-s + 8.31·9-s − 1.70·10-s − 5.80·11-s + 13.2·12-s − 5.40·13-s − 5.61·14-s + 2.34·15-s + 3.59·16-s + 0.263·17-s − 20.2·18-s + 1.83·19-s + 2.74·20-s + 7.75·21-s + 14.1·22-s + 8.68·23-s − 15.8·24-s − 4.51·25-s + 13.1·26-s + 17.8·27-s + 9.06·28-s + ⋯ |
| L(s) = 1 | − 1.72·2-s + 1.94·3-s + 1.96·4-s + 0.312·5-s − 3.34·6-s + 0.871·7-s − 1.66·8-s + 2.77·9-s − 0.537·10-s − 1.75·11-s + 3.81·12-s − 1.49·13-s − 1.50·14-s + 0.606·15-s + 0.899·16-s + 0.0638·17-s − 4.77·18-s + 0.419·19-s + 0.613·20-s + 1.69·21-s + 3.01·22-s + 1.81·23-s − 3.23·24-s − 0.902·25-s + 2.58·26-s + 3.44·27-s + 1.71·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5077 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5077 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.055223533\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.055223533\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5077 | \( 1+O(T) \) |
| good | 2 | \( 1 + 2.43T + 2T^{2} \) |
| 3 | \( 1 - 3.36T + 3T^{2} \) |
| 5 | \( 1 - 0.698T + 5T^{2} \) |
| 7 | \( 1 - 2.30T + 7T^{2} \) |
| 11 | \( 1 + 5.80T + 11T^{2} \) |
| 13 | \( 1 + 5.40T + 13T^{2} \) |
| 17 | \( 1 - 0.263T + 17T^{2} \) |
| 19 | \( 1 - 1.83T + 19T^{2} \) |
| 23 | \( 1 - 8.68T + 23T^{2} \) |
| 29 | \( 1 - 4.98T + 29T^{2} \) |
| 31 | \( 1 - 8.70T + 31T^{2} \) |
| 37 | \( 1 - 4.86T + 37T^{2} \) |
| 41 | \( 1 + 2.49T + 41T^{2} \) |
| 43 | \( 1 + 8.28T + 43T^{2} \) |
| 47 | \( 1 - 12.0T + 47T^{2} \) |
| 53 | \( 1 + 3.11T + 53T^{2} \) |
| 59 | \( 1 - 8.14T + 59T^{2} \) |
| 61 | \( 1 + 9.99T + 61T^{2} \) |
| 67 | \( 1 - 9.95T + 67T^{2} \) |
| 71 | \( 1 - 3.63T + 71T^{2} \) |
| 73 | \( 1 - 6.71T + 73T^{2} \) |
| 79 | \( 1 + 4.60T + 79T^{2} \) |
| 83 | \( 1 + 3.46T + 83T^{2} \) |
| 89 | \( 1 - 6.50T + 89T^{2} \) |
| 97 | \( 1 + 16.3T + 97T^{2} \) |
| 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
−8.235266529666900708095830179895, −7.80818740438735494694815265671, −7.38696660018115481766969893992, −6.71601471010208048789859698021, −5.13662897102235378324493933793, −4.57386997299086453417127734437, −2.99691421970345258982418729121, −2.58839583737706568882692768033, −1.98940851453697359451902690068, −0.934870558188759842055373103064,
0.934870558188759842055373103064, 1.98940851453697359451902690068, 2.58839583737706568882692768033, 2.99691421970345258982418729121, 4.57386997299086453417127734437, 5.13662897102235378324493933793, 6.71601471010208048789859698021, 7.38696660018115481766969893992, 7.80818740438735494694815265671, 8.235266529666900708095830179895
