L(s) = 1 | + (−1.79 + 1.79i)3-s + (3.30 − 3.30i)7-s − 3.44i·9-s + 2.44·11-s + (−2.60 + 2.60i)13-s + (1.48 − 1.48i)17-s + (2.66 + 3.44i)19-s + 11.8i·21-s + (−4.78 − 4.78i)23-s + (0.807 + 0.807i)27-s + 5.32·29-s − 6.52i·31-s + (−4.39 + 4.39i)33-s + (−7.00 − 7.00i)37-s − 9.34i·39-s + ⋯ |
L(s) = 1 | + (−1.03 + 1.03i)3-s + (1.24 − 1.24i)7-s − 1.14i·9-s + 0.738·11-s + (−0.721 + 0.721i)13-s + (0.359 − 0.359i)17-s + (0.611 + 0.791i)19-s + 2.58i·21-s + (−0.997 − 0.997i)23-s + (0.155 + 0.155i)27-s + 0.989·29-s − 1.17i·31-s + (−0.765 + 0.765i)33-s + (−1.15 − 1.15i)37-s − 1.49i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.936 + 0.351i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.936 + 0.351i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.268942200\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.268942200\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (-2.66 - 3.44i)T \) |
good | 3 | \( 1 + (1.79 - 1.79i)T - 3iT^{2} \) |
| 7 | \( 1 + (-3.30 + 3.30i)T - 7iT^{2} \) |
| 11 | \( 1 - 2.44T + 11T^{2} \) |
| 13 | \( 1 + (2.60 - 2.60i)T - 13iT^{2} \) |
| 17 | \( 1 + (-1.48 + 1.48i)T - 17iT^{2} \) |
| 23 | \( 1 + (4.78 + 4.78i)T + 23iT^{2} \) |
| 29 | \( 1 - 5.32T + 29T^{2} \) |
| 31 | \( 1 + 6.52iT - 31T^{2} \) |
| 37 | \( 1 + (7.00 + 7.00i)T + 37iT^{2} \) |
| 41 | \( 1 + 11.8iT - 41T^{2} \) |
| 43 | \( 1 + (-1.81 - 1.81i)T + 43iT^{2} \) |
| 47 | \( 1 + (3.30 - 3.30i)T - 47iT^{2} \) |
| 53 | \( 1 + (-3.41 + 3.41i)T - 53iT^{2} \) |
| 59 | \( 1 + 6.52T + 59T^{2} \) |
| 61 | \( 1 - 5.34T + 61T^{2} \) |
| 67 | \( 1 + (-4.58 - 4.58i)T + 67iT^{2} \) |
| 71 | \( 1 - 11.8iT - 71T^{2} \) |
| 73 | \( 1 + (-2.96 - 2.96i)T + 73iT^{2} \) |
| 79 | \( 1 - 11.8T + 79T^{2} \) |
| 83 | \( 1 + (6.93 + 6.93i)T + 83iT^{2} \) |
| 89 | \( 1 - 5.32T + 89T^{2} \) |
| 97 | \( 1 + (3.77 + 3.77i)T + 97iT^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.450921094163226448307126615829, −8.381687950653090649153882986102, −7.51686636332300984232925082672, −6.79817412865042063817226005082, −5.75402742144541132753615963031, −4.99803602178209119728158484998, −4.22605110429191507559252344850, −3.86947751001181994531498702352, −1.97908097687730032188897145907, −0.62200325344866657122162414351,
1.15264256493763482744146732662, 1.94573386090091691086517695154, 3.17299459839491707542626766373, 4.86330672548133135333698969010, 5.23186723215148171015351983936, 6.07696555229895532708039902128, 6.77825690196074310282002319318, 7.73264030851622644166428458663, 8.235636249652162090356386425366, 9.158902620925799464663912864521