| L(s) = 1 | − i·3-s + 4.62i·7-s − 9-s − 4.94·11-s + 3.76i·13-s − 2.69i·17-s − 5.87·19-s + 4.62·21-s − 6.67i·23-s + i·27-s − 1.20·29-s + 3.30·31-s + 4.94i·33-s + 1.87i·37-s + 3.76·39-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + 1.74i·7-s − 0.333·9-s − 1.49·11-s + 1.04i·13-s − 0.652i·17-s − 1.34·19-s + 1.00·21-s − 1.39i·23-s + 0.192i·27-s − 0.222·29-s + 0.593·31-s + 0.861i·33-s + 0.308i·37-s + 0.602·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7500 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6646619437\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6646619437\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 4.62iT - 7T^{2} \) |
| 11 | \( 1 + 4.94T + 11T^{2} \) |
| 13 | \( 1 - 3.76iT - 13T^{2} \) |
| 17 | \( 1 + 2.69iT - 17T^{2} \) |
| 19 | \( 1 + 5.87T + 19T^{2} \) |
| 23 | \( 1 + 6.67iT - 23T^{2} \) |
| 29 | \( 1 + 1.20T + 29T^{2} \) |
| 31 | \( 1 - 3.30T + 31T^{2} \) |
| 37 | \( 1 - 1.87iT - 37T^{2} \) |
| 41 | \( 1 + 3.03T + 41T^{2} \) |
| 43 | \( 1 - 10.6iT - 43T^{2} \) |
| 47 | \( 1 - 0.259iT - 47T^{2} \) |
| 53 | \( 1 + 9.79iT - 53T^{2} \) |
| 59 | \( 1 - 9.62T + 59T^{2} \) |
| 61 | \( 1 - 6.27T + 61T^{2} \) |
| 67 | \( 1 - 2.56iT - 67T^{2} \) |
| 71 | \( 1 - 8.67T + 71T^{2} \) |
| 73 | \( 1 + 4.87iT - 73T^{2} \) |
| 79 | \( 1 + 12.4T + 79T^{2} \) |
| 83 | \( 1 - 8.89iT - 83T^{2} \) |
| 89 | \( 1 + 14.5T + 89T^{2} \) |
| 97 | \( 1 + 3.98iT - 97T^{2} \) |
| 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
−7.908253268129509321227975664555, −6.79395444686057042122698538452, −6.45833455683089795904098245477, −5.61069222318131269484406387411, −5.03476634018880923383127024488, −4.30519747553957828972202597832, −2.88181218503204692698926816105, −2.48390609795505354265973111265, −1.83784929863262003188159887173, −0.19794357523730295433821033938,
0.78019200639075711955904668576, 2.07783999731006941599601689743, 3.11773512869435415162182997579, 3.81724441969642739059576236588, 4.38420629612341942075927067590, 5.27682309801609288903072659019, 5.77332812428517397683117966413, 6.82299035613047145566108816651, 7.43870097850522001241916583320, 8.057548537922802439347955269900
