L(s) = 1 | + (−4.37 + 2.52i)2-s + (5.16 + 0.559i)3-s + (8.77 − 15.1i)4-s + (−3.07 − 10.7i)5-s + (−24.0 + 10.6i)6-s + (18.1 − 10.5i)7-s + 48.2i·8-s + (26.3 + 5.78i)9-s + (40.6 + 39.2i)10-s + (14.2 + 24.7i)11-s + (53.8 − 73.5i)12-s + (−8.72 − 5.03i)13-s + (−53.1 + 91.9i)14-s + (−9.86 − 57.2i)15-s + (−51.7 − 89.6i)16-s − 82.7i·17-s + ⋯ |
L(s) = 1 | + (−1.54 + 0.893i)2-s + (0.994 + 0.107i)3-s + (1.09 − 1.89i)4-s + (−0.274 − 0.961i)5-s + (−1.63 + 0.721i)6-s + (0.982 − 0.567i)7-s + 2.13i·8-s + (0.976 + 0.214i)9-s + (1.28 + 1.24i)10-s + (0.391 + 0.678i)11-s + (1.29 − 1.77i)12-s + (−0.186 − 0.107i)13-s + (−1.01 + 1.75i)14-s + (−0.169 − 0.985i)15-s + (−0.808 − 1.40i)16-s − 1.18i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 - 0.235i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.971 - 0.235i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.931515 + 0.111037i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.931515 + 0.111037i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-5.16 - 0.559i)T \) |
| 5 | \( 1 + (3.07 + 10.7i)T \) |
good | 2 | \( 1 + (4.37 - 2.52i)T + (4 - 6.92i)T^{2} \) |
| 7 | \( 1 + (-18.1 + 10.5i)T + (171.5 - 297. i)T^{2} \) |
| 11 | \( 1 + (-14.2 - 24.7i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + (8.72 + 5.03i)T + (1.09e3 + 1.90e3i)T^{2} \) |
| 17 | \( 1 + 82.7iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 1.91T + 6.85e3T^{2} \) |
| 23 | \( 1 + (-147. - 85.2i)T + (6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (128. + 222. i)T + (-1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 + (24.0 - 41.6i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 - 161. iT - 5.06e4T^{2} \) |
| 41 | \( 1 + (139. - 241. i)T + (-3.44e4 - 5.96e4i)T^{2} \) |
| 43 | \( 1 + (233. - 134. i)T + (3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 + (8.30 - 4.79i)T + (5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + 35.7iT - 1.48e5T^{2} \) |
| 59 | \( 1 + (281. - 487. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-39.6 - 68.6i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (404. + 233. i)T + (1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 - 316.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 633. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + (395. + 685. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (197. - 114. i)T + (2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 + 53.9T + 7.04e5T^{2} \) |
| 97 | \( 1 + (83.7 - 48.3i)T + (4.56e5 - 7.90e5i)T^{2} \) |
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
−15.49349324606336025228444004018, −14.77666259140167627006976629698, −13.41628215194485252714799471205, −11.47194102844799984661658563814, −9.858191985058069923215373536071, −9.062615797259890767323729120588, −7.958432984064842884209053735917, −7.23978934750438407042717219405, −4.79947611430918987878509760131, −1.36650640148180238890466420988,
1.89941574549633252367207202499, 3.37222849247477998789449646729, 7.08636762711157414377615321310, 8.278055290616344394234330775528, 9.002001542131825022276991382142, 10.48766435245690953563098008852, 11.28400896395939299761280854406, 12.57480136985567176983799462363, 14.37560777165430249504754014483, 15.23009473389638818428303140810