| L(s) = 1 | − 2-s − 0.340·3-s + 4-s + 1.77·5-s + 0.340·6-s + 4.18·7-s − 8-s − 2.88·9-s − 1.77·10-s + 2.61·11-s − 0.340·12-s + 3.00·13-s − 4.18·14-s − 0.603·15-s + 16-s + 3.51·17-s + 2.88·18-s + 7.65·19-s + 1.77·20-s − 1.42·21-s − 2.61·22-s + 3.71·23-s + 0.340·24-s − 1.86·25-s − 3.00·26-s + 2.00·27-s + 4.18·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.196·3-s + 0.5·4-s + 0.791·5-s + 0.139·6-s + 1.58·7-s − 0.353·8-s − 0.961·9-s − 0.559·10-s + 0.787·11-s − 0.0983·12-s + 0.834·13-s − 1.11·14-s − 0.155·15-s + 0.250·16-s + 0.852·17-s + 0.679·18-s + 1.75·19-s + 0.395·20-s − 0.311·21-s − 0.557·22-s + 0.775·23-s + 0.0695·24-s − 0.372·25-s − 0.590·26-s + 0.385·27-s + 0.790·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.118322226\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.118322226\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 2003 | \( 1 - T \) |
| good | 3 | \( 1 + 0.340T + 3T^{2} \) |
| 5 | \( 1 - 1.77T + 5T^{2} \) |
| 7 | \( 1 - 4.18T + 7T^{2} \) |
| 11 | \( 1 - 2.61T + 11T^{2} \) |
| 13 | \( 1 - 3.00T + 13T^{2} \) |
| 17 | \( 1 - 3.51T + 17T^{2} \) |
| 19 | \( 1 - 7.65T + 19T^{2} \) |
| 23 | \( 1 - 3.71T + 23T^{2} \) |
| 29 | \( 1 + 3.40T + 29T^{2} \) |
| 31 | \( 1 - 2.86T + 31T^{2} \) |
| 37 | \( 1 + 2.37T + 37T^{2} \) |
| 41 | \( 1 + 3.10T + 41T^{2} \) |
| 43 | \( 1 + 0.620T + 43T^{2} \) |
| 47 | \( 1 - 2.52T + 47T^{2} \) |
| 53 | \( 1 - 10.7T + 53T^{2} \) |
| 59 | \( 1 - 10.6T + 59T^{2} \) |
| 61 | \( 1 + 12.7T + 61T^{2} \) |
| 67 | \( 1 + 16.2T + 67T^{2} \) |
| 71 | \( 1 - 8.12T + 71T^{2} \) |
| 73 | \( 1 + 11.4T + 73T^{2} \) |
| 79 | \( 1 + 12.7T + 79T^{2} \) |
| 83 | \( 1 - 4.18T + 83T^{2} \) |
| 89 | \( 1 + 7.88T + 89T^{2} \) |
| 97 | \( 1 - 6.46T + 97T^{2} \) |
| 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
−8.656135612968270045751373260907, −7.75159543351766050372082710735, −7.21532415026305865354917079171, −6.08710401094273006916317800446, −5.59596603296590352165681192086, −4.97862598008271822663332396811, −3.71533441701214962190191718136, −2.76344779312551324726647623660, −1.59470646005161829202512810314, −1.08232378063886222204190356517,
1.08232378063886222204190356517, 1.59470646005161829202512810314, 2.76344779312551324726647623660, 3.71533441701214962190191718136, 4.97862598008271822663332396811, 5.59596603296590352165681192086, 6.08710401094273006916317800446, 7.21532415026305865354917079171, 7.75159543351766050372082710735, 8.656135612968270045751373260907
