| L(s) = 1 | + (−0.567 + 2.77i)2-s + (−5.84 + 3.12i)3-s + (−7.35 − 3.14i)4-s + (0.807 + 0.983i)5-s + (−5.33 − 17.9i)6-s + (−28.4 + 18.9i)7-s + (12.8 − 18.5i)8-s + (9.36 − 14.0i)9-s + (−3.18 + 1.67i)10-s + (45.4 − 13.7i)11-s + (52.7 − 4.59i)12-s + (−48.6 − 39.9i)13-s + (−36.5 − 89.5i)14-s + (−7.78 − 3.22i)15-s + (44.2 + 46.2i)16-s + (64.3 − 26.6i)17-s + ⋯ |
| L(s) = 1 | + (−0.200 + 0.979i)2-s + (−1.12 + 0.600i)3-s + (−0.919 − 0.393i)4-s + (0.0722 + 0.0879i)5-s + (−0.363 − 1.22i)6-s + (−1.53 + 1.02i)7-s + (0.569 − 0.821i)8-s + (0.346 − 0.519i)9-s + (−0.100 + 0.0530i)10-s + (1.24 − 0.377i)11-s + (1.26 − 0.110i)12-s + (−1.03 − 0.851i)13-s + (−0.697 − 1.70i)14-s + (−0.134 − 0.0555i)15-s + (0.690 + 0.722i)16-s + (0.918 − 0.380i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.836 + 0.547i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.836 + 0.547i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.183061 - 0.0545711i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.183061 - 0.0545711i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.567 - 2.77i)T \) |
| good | 3 | \( 1 + (5.84 - 3.12i)T + (15.0 - 22.4i)T^{2} \) |
| 5 | \( 1 + (-0.807 - 0.983i)T + (-24.3 + 122. i)T^{2} \) |
| 7 | \( 1 + (28.4 - 18.9i)T + (131. - 316. i)T^{2} \) |
| 11 | \( 1 + (-45.4 + 13.7i)T + (1.10e3 - 739. i)T^{2} \) |
| 13 | \( 1 + (48.6 + 39.9i)T + (428. + 2.15e3i)T^{2} \) |
| 17 | \( 1 + (-64.3 + 26.6i)T + (3.47e3 - 3.47e3i)T^{2} \) |
| 19 | \( 1 + (-13.3 - 135. i)T + (-6.72e3 + 1.33e3i)T^{2} \) |
| 23 | \( 1 + (-19.8 + 3.95i)T + (1.12e4 - 4.65e3i)T^{2} \) |
| 29 | \( 1 + (24.6 - 81.2i)T + (-2.02e4 - 1.35e4i)T^{2} \) |
| 31 | \( 1 + (73.2 + 73.2i)T + 2.97e4iT^{2} \) |
| 37 | \( 1 + (127. + 12.5i)T + (4.96e4 + 9.88e3i)T^{2} \) |
| 41 | \( 1 + (73.2 + 368. i)T + (-6.36e4 + 2.63e4i)T^{2} \) |
| 43 | \( 1 + (386. + 206. i)T + (4.41e4 + 6.61e4i)T^{2} \) |
| 47 | \( 1 + (140. + 339. i)T + (-7.34e4 + 7.34e4i)T^{2} \) |
| 53 | \( 1 + (-128. - 422. i)T + (-1.23e5 + 8.27e4i)T^{2} \) |
| 59 | \( 1 + (16.9 - 13.8i)T + (4.00e4 - 2.01e5i)T^{2} \) |
| 61 | \( 1 + (136. + 254. i)T + (-1.26e5 + 1.88e5i)T^{2} \) |
| 67 | \( 1 + (-349. - 653. i)T + (-1.67e5 + 2.50e5i)T^{2} \) |
| 71 | \( 1 + (-32.1 - 48.1i)T + (-1.36e5 + 3.30e5i)T^{2} \) |
| 73 | \( 1 + (-148. - 99.4i)T + (1.48e5 + 3.59e5i)T^{2} \) |
| 79 | \( 1 + (-66.9 + 161. i)T + (-3.48e5 - 3.48e5i)T^{2} \) |
| 83 | \( 1 + (1.25e3 - 123. i)T + (5.60e5 - 1.11e5i)T^{2} \) |
| 89 | \( 1 + (-350. - 69.6i)T + (6.51e5 + 2.69e5i)T^{2} \) |
| 97 | \( 1 + (533. + 533. i)T + 9.12e5iT^{2} \) |
| 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
−12.50332782693348291740658855950, −12.02875770340753940001794913095, −10.19854166498352157837674640689, −9.808169510538486976792535029446, −8.604929686209321517138791510200, −6.95687163291693886673944892434, −5.90630843762402536086549330395, −5.38965324612109579143774519075, −3.60273599454953878518291125856, −0.14086606704817761026365495829,
1.20608771388166274230706738817, 3.38635050004815061722647802273, 4.81365165497114795843595671544, 6.53625467421804256672926993397, 7.22009803593463518535105325616, 9.312922814919499038936106398627, 9.842219820041002527710786171089, 11.13647366863753964387009425648, 11.90839111586209618415923627155, 12.74205838500324976595662626040
