L(s) = 1 | + (0.766 − 0.642i)3-s + (3.20 + 1.16i)5-s + (−1.43 + 2.49i)7-s + (0.173 − 0.984i)9-s + (−0.173 − 0.300i)11-s + (−1.26 − 1.06i)13-s + (3.20 − 1.16i)15-s + (1.20 + 6.83i)17-s + (2.82 + 3.31i)19-s + (0.500 + 2.83i)21-s + (6.39 − 2.32i)23-s + (5.08 + 4.26i)25-s + (−0.500 − 0.866i)27-s + (1.10 − 6.25i)29-s + (0.798 − 1.38i)31-s + ⋯ |
L(s) = 1 | + (0.442 − 0.371i)3-s + (1.43 + 0.521i)5-s + (−0.544 + 0.942i)7-s + (0.0578 − 0.328i)9-s + (−0.0523 − 0.0906i)11-s + (−0.351 − 0.294i)13-s + (0.827 − 0.301i)15-s + (0.292 + 1.65i)17-s + (0.648 + 0.761i)19-s + (0.109 + 0.618i)21-s + (1.33 − 0.484i)23-s + (1.01 + 0.853i)25-s + (−0.0962 − 0.166i)27-s + (0.204 − 1.16i)29-s + (0.143 − 0.248i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.854 - 0.519i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.854 - 0.519i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.13431 + 0.598468i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.13431 + 0.598468i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.766 + 0.642i)T \) |
| 19 | \( 1 + (-2.82 - 3.31i)T \) |
good | 5 | \( 1 + (-3.20 - 1.16i)T + (3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (1.43 - 2.49i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (0.173 + 0.300i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (1.26 + 1.06i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-1.20 - 6.83i)T + (-15.9 + 5.81i)T^{2} \) |
| 23 | \( 1 + (-6.39 + 2.32i)T + (17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (-1.10 + 6.25i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (-0.798 + 1.38i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 11.2T + 37T^{2} \) |
| 41 | \( 1 + (2.67 - 2.24i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-2.14 - 0.780i)T + (32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (0.971 - 5.51i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + (1.86 - 0.677i)T + (40.6 - 34.0i)T^{2} \) |
| 59 | \( 1 + (0.0773 + 0.438i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (-11.7 + 4.28i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (-0.187 + 1.06i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (15.6 + 5.68i)T + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (-9.51 + 7.98i)T + (12.6 - 71.8i)T^{2} \) |
| 79 | \( 1 + (-8.36 + 7.02i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-5.85 + 10.1i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-1.37 - 1.15i)T + (15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (-0.634 - 3.59i)T + (-91.1 + 33.1i)T^{2} \) |
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
−10.08239747850736511663948037412, −9.364767229277637071762747060947, −8.620730149970394350592167151585, −7.66854237898535540541020139012, −6.45151475870535825102744135379, −6.06322426153417502958103735697, −5.17185111262734469171034311758, −3.45078067921121123017215990441, −2.58863457939569923675078311972, −1.65877458267209226100391789598,
1.10194035797027313256346712363, 2.53965684297629802787340502179, 3.51761039911733996323661124083, 5.04442535908172923999168695749, 5.24281585825146109694561140762, 6.94367889096208036666472932170, 7.11240017665034013858180436653, 8.724520040812808692155603067637, 9.294931415928254051018939433607, 9.884992226634945096710460414892