L(s) = 1 | + 2-s − 2.20·3-s + 4-s + 2.72·5-s − 2.20·6-s + 3.76·7-s + 8-s + 1.85·9-s + 2.72·10-s − 6.14·11-s − 2.20·12-s + 5.77·13-s + 3.76·14-s − 6.00·15-s + 16-s − 8.03·17-s + 1.85·18-s + 0.163·19-s + 2.72·20-s − 8.29·21-s − 6.14·22-s + 8.68·23-s − 2.20·24-s + 2.42·25-s + 5.77·26-s + 2.52·27-s + 3.76·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.27·3-s + 0.5·4-s + 1.21·5-s − 0.899·6-s + 1.42·7-s + 0.353·8-s + 0.617·9-s + 0.861·10-s − 1.85·11-s − 0.635·12-s + 1.60·13-s + 1.00·14-s − 1.55·15-s + 0.250·16-s − 1.94·17-s + 0.436·18-s + 0.0373·19-s + 0.609·20-s − 1.80·21-s − 1.30·22-s + 1.81·23-s − 0.449·24-s + 0.485·25-s + 1.13·26-s + 0.486·27-s + 0.711·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.790223238\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.790223238\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 2011 | \( 1 + T \) |
good | 3 | \( 1 + 2.20T + 3T^{2} \) |
| 5 | \( 1 - 2.72T + 5T^{2} \) |
| 7 | \( 1 - 3.76T + 7T^{2} \) |
| 11 | \( 1 + 6.14T + 11T^{2} \) |
| 13 | \( 1 - 5.77T + 13T^{2} \) |
| 17 | \( 1 + 8.03T + 17T^{2} \) |
| 19 | \( 1 - 0.163T + 19T^{2} \) |
| 23 | \( 1 - 8.68T + 23T^{2} \) |
| 29 | \( 1 + 8.02T + 29T^{2} \) |
| 31 | \( 1 - 9.30T + 31T^{2} \) |
| 37 | \( 1 + 2.01T + 37T^{2} \) |
| 41 | \( 1 - 10.4T + 41T^{2} \) |
| 43 | \( 1 - 8.21T + 43T^{2} \) |
| 47 | \( 1 + 6.75T + 47T^{2} \) |
| 53 | \( 1 - 13.2T + 53T^{2} \) |
| 59 | \( 1 - 2.35T + 59T^{2} \) |
| 61 | \( 1 + 3.06T + 61T^{2} \) |
| 67 | \( 1 + 1.00T + 67T^{2} \) |
| 71 | \( 1 - 3.15T + 71T^{2} \) |
| 73 | \( 1 + 3.71T + 73T^{2} \) |
| 79 | \( 1 - 1.57T + 79T^{2} \) |
| 83 | \( 1 + 14.6T + 83T^{2} \) |
| 89 | \( 1 - 12.8T + 89T^{2} \) |
| 97 | \( 1 + 1.09T + 97T^{2} \) |
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
−8.450750193759787010953090704482, −7.51417192158194711037624093542, −6.66179597786820229042863893707, −5.91247386678810342646246480392, −5.50840163054701408628957052285, −4.89752918406878265830132192385, −4.31085757787914504066977816859, −2.75015678866112064534239566499, −2.02391391247712372704025524563, −0.951049137582017911525853198483,
0.951049137582017911525853198483, 2.02391391247712372704025524563, 2.75015678866112064534239566499, 4.31085757787914504066977816859, 4.89752918406878265830132192385, 5.50840163054701408628957052285, 5.91247386678810342646246480392, 6.66179597786820229042863893707, 7.51417192158194711037624093542, 8.450750193759787010953090704482