L(s) = 1 | + (−0.452 − 5.17i)3-s + (6.54 + 11.3i)5-s + (1.26 + 18.4i)7-s + (−26.5 + 4.68i)9-s + (−45.6 − 26.3i)11-s + (−31.8 − 18.3i)13-s + (55.7 − 39.0i)15-s + (−20.4 − 35.4i)17-s + (−108. − 62.8i)19-s + (95.0 − 14.9i)21-s + (1.99 − 1.14i)23-s + (−23.1 + 40.0i)25-s + (36.3 + 135. i)27-s + (−151. + 87.5i)29-s − 268. i·31-s + ⋯ |
L(s) = 1 | + (−0.0871 − 0.996i)3-s + (0.585 + 1.01i)5-s + (0.0685 + 0.997i)7-s + (−0.984 + 0.173i)9-s + (−1.25 − 0.722i)11-s + (−0.679 − 0.392i)13-s + (0.958 − 0.671i)15-s + (−0.292 − 0.506i)17-s + (−1.31 − 0.758i)19-s + (0.987 − 0.155i)21-s + (0.0180 − 0.0104i)23-s + (−0.185 + 0.320i)25-s + (0.258 + 0.965i)27-s + (−0.971 + 0.560i)29-s − 1.55i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 - 0.230i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.973 - 0.230i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.02741197915\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.02741197915\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.452 + 5.17i)T \) |
| 7 | \( 1 + (-1.26 - 18.4i)T \) |
good | 5 | \( 1 + (-6.54 - 11.3i)T + (-62.5 + 108. i)T^{2} \) |
| 11 | \( 1 + (45.6 + 26.3i)T + (665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + (31.8 + 18.3i)T + (1.09e3 + 1.90e3i)T^{2} \) |
| 17 | \( 1 + (20.4 + 35.4i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (108. + 62.8i)T + (3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-1.99 + 1.14i)T + (6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + (151. - 87.5i)T + (1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + 268. iT - 2.97e4T^{2} \) |
| 37 | \( 1 + (191. - 331. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + (75.5 - 130. i)T + (-3.44e4 - 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-23.2 - 40.2i)T + (-3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 - 372.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (-539. + 311. i)T + (7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 - 4.01T + 2.05e5T^{2} \) |
| 61 | \( 1 - 779. iT - 2.26e5T^{2} \) |
| 67 | \( 1 + 439.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 800. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (428. - 247. i)T + (1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 - 140.T + 4.93e5T^{2} \) |
| 83 | \( 1 + (-506. - 877. i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 + (-211. + 366. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + (264. - 152. i)T + (4.56e5 - 7.90e5i)T^{2} \) |
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
−11.12556620374719359883724372608, −10.34246367292800615651932848310, −8.993112582564228074402794416188, −8.042157476664408843002275548795, −6.99937725240066135304310495585, −6.05754852136654496641300118936, −5.23381508102808358086525033671, −2.79055867596661221316884143080, −2.31684873136777361714918505188, −0.009785305991716759405084855363,
2.04840004948245426138634126182, 3.95742660677340239554018482206, 4.81081481060909512541324722499, 5.68644195436119016399809687811, 7.23774081314026179847187889174, 8.448161901745808939855531963887, 9.330231184371446114853824395170, 10.36232026311370480949058694786, 10.67614602727384357119441858383, 12.25778590519389704316212744435