| L(s) = 1 | − 1.78·2-s + 3-s + 1.18·4-s + 3.42·5-s − 1.78·6-s + 3.39·7-s + 1.45·8-s + 9-s − 6.10·10-s + 1.48·11-s + 1.18·12-s + 1.54·13-s − 6.06·14-s + 3.42·15-s − 4.96·16-s + 1.71·17-s − 1.78·18-s + 5.94·19-s + 4.04·20-s + 3.39·21-s − 2.65·22-s − 23-s + 1.45·24-s + 6.71·25-s − 2.74·26-s + 27-s + 4.01·28-s + ⋯ |
| L(s) = 1 | − 1.26·2-s + 0.577·3-s + 0.591·4-s + 1.53·5-s − 0.728·6-s + 1.28·7-s + 0.515·8-s + 0.333·9-s − 1.93·10-s + 0.448·11-s + 0.341·12-s + 0.427·13-s − 1.62·14-s + 0.883·15-s − 1.24·16-s + 0.415·17-s − 0.420·18-s + 1.36·19-s + 0.905·20-s + 0.741·21-s − 0.566·22-s − 0.208·23-s + 0.297·24-s + 1.34·25-s − 0.539·26-s + 0.192·27-s + 0.759·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.926333768\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.926333768\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 + 1.78T + 2T^{2} \) |
| 5 | \( 1 - 3.42T + 5T^{2} \) |
| 7 | \( 1 - 3.39T + 7T^{2} \) |
| 11 | \( 1 - 1.48T + 11T^{2} \) |
| 13 | \( 1 - 1.54T + 13T^{2} \) |
| 17 | \( 1 - 1.71T + 17T^{2} \) |
| 19 | \( 1 - 5.94T + 19T^{2} \) |
| 31 | \( 1 + 2.03T + 31T^{2} \) |
| 37 | \( 1 + 2.86T + 37T^{2} \) |
| 41 | \( 1 - 2.74T + 41T^{2} \) |
| 43 | \( 1 - 2.40T + 43T^{2} \) |
| 47 | \( 1 - 0.626T + 47T^{2} \) |
| 53 | \( 1 + 1.96T + 53T^{2} \) |
| 59 | \( 1 + 10.3T + 59T^{2} \) |
| 61 | \( 1 + 6.34T + 61T^{2} \) |
| 67 | \( 1 + 3.34T + 67T^{2} \) |
| 71 | \( 1 + 15.8T + 71T^{2} \) |
| 73 | \( 1 + 8.02T + 73T^{2} \) |
| 79 | \( 1 - 2.84T + 79T^{2} \) |
| 83 | \( 1 + 16.8T + 83T^{2} \) |
| 89 | \( 1 + 14.5T + 89T^{2} \) |
| 97 | \( 1 - 6.93T + 97T^{2} \) |
| 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
−9.137217432152482213306195327674, −8.619427449384266510539120392440, −7.75359987885965335971507007544, −7.21399473496974284347795482145, −6.04558653983551340748876279818, −5.24963834774676806877310397326, −4.30640666727410919783323574879, −2.86834505910030160111158488326, −1.63163062873299282224089069959, −1.38037479549040105693073579085,
1.38037479549040105693073579085, 1.63163062873299282224089069959, 2.86834505910030160111158488326, 4.30640666727410919783323574879, 5.24963834774676806877310397326, 6.04558653983551340748876279818, 7.21399473496974284347795482145, 7.75359987885965335971507007544, 8.619427449384266510539120392440, 9.137217432152482213306195327674
