L(s) = 1 | + 0.568i·3-s + (−2.03 − 0.918i)5-s + (1.76 − 1.96i)7-s + 2.67·9-s − 0.417i·11-s − 2.13·13-s + (0.521 − 1.15i)15-s + 0.314·17-s + 2.19·19-s + (1.11 + 1.00i)21-s + 0.992·23-s + (3.31 + 3.74i)25-s + 3.22i·27-s + 4.35·29-s − 4.22·31-s + ⋯ |
L(s) = 1 | + 0.328i·3-s + (−0.911 − 0.410i)5-s + (0.667 − 0.744i)7-s + 0.892·9-s − 0.125i·11-s − 0.592·13-s + (0.134 − 0.299i)15-s + 0.0761·17-s + 0.503·19-s + (0.244 + 0.219i)21-s + 0.206·23-s + (0.662 + 0.748i)25-s + 0.620i·27-s + 0.808·29-s − 0.759·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.404 + 0.914i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.404 + 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.522574434\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.522574434\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (2.03 + 0.918i)T \) |
| 7 | \( 1 + (-1.76 + 1.96i)T \) |
good | 3 | \( 1 - 0.568iT - 3T^{2} \) |
| 11 | \( 1 + 0.417iT - 11T^{2} \) |
| 13 | \( 1 + 2.13T + 13T^{2} \) |
| 17 | \( 1 - 0.314T + 17T^{2} \) |
| 19 | \( 1 - 2.19T + 19T^{2} \) |
| 23 | \( 1 - 0.992T + 23T^{2} \) |
| 29 | \( 1 - 4.35T + 29T^{2} \) |
| 31 | \( 1 + 4.22T + 31T^{2} \) |
| 37 | \( 1 + 6.83iT - 37T^{2} \) |
| 41 | \( 1 + 3.82iT - 41T^{2} \) |
| 43 | \( 1 + 6.04T + 43T^{2} \) |
| 47 | \( 1 + 4.98iT - 47T^{2} \) |
| 53 | \( 1 + 7.14iT - 53T^{2} \) |
| 59 | \( 1 - 9.50T + 59T^{2} \) |
| 61 | \( 1 - 0.783iT - 61T^{2} \) |
| 67 | \( 1 + 11.1T + 67T^{2} \) |
| 71 | \( 1 + 6.55iT - 71T^{2} \) |
| 73 | \( 1 - 5.11T + 73T^{2} \) |
| 79 | \( 1 - 11.4iT - 79T^{2} \) |
| 83 | \( 1 - 5.49iT - 83T^{2} \) |
| 89 | \( 1 + 12.5iT - 89T^{2} \) |
| 97 | \( 1 - 0.314T + 97T^{2} \) |
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
−8.806197203249286887655214843657, −8.093715278941595851961659476068, −7.27808278105533659943411816487, −6.97156000034754401343778214987, −5.43407275140327250698032584292, −4.76599023193408492098790560049, −4.08675550367761087253712729641, −3.35523530458523527285834889002, −1.79499176491672630894768149695, −0.60489524740635845963814219424,
1.20568568108579878526597634021, 2.38885072042589752116138004568, 3.34477242438809958903381319199, 4.47946138819393759734280030070, 5.00595301018308528878321632314, 6.19119776776137111427772382800, 7.01636981423233697506604561031, 7.63967108606605981105216442331, 8.217327004435394981232244528006, 9.085780126171259996871904140038