L(s) = 1 | + (0.521 + 0.654i)2-s + (5.78 − 2.78i)3-s + (1.62 − 7.11i)4-s + (−1.28 + 0.620i)5-s + (4.83 + 2.32i)6-s + (−17.3 − 6.46i)7-s + (11.5 − 5.55i)8-s + (8.82 − 11.0i)9-s + (−1.07 − 0.519i)10-s + (33.8 + 42.5i)11-s + (−10.4 − 45.6i)12-s + (41.8 + 52.5i)13-s + (−4.82 − 14.7i)14-s + (−5.72 + 7.17i)15-s + (−42.9 − 20.6i)16-s + (−4.21 − 18.4i)17-s + ⋯ |
L(s) = 1 | + (0.184 + 0.231i)2-s + (1.11 − 0.535i)3-s + (0.203 − 0.889i)4-s + (−0.115 + 0.0555i)5-s + (0.329 + 0.158i)6-s + (−0.937 − 0.349i)7-s + (0.509 − 0.245i)8-s + (0.326 − 0.410i)9-s + (−0.0341 − 0.0164i)10-s + (0.929 + 1.16i)11-s + (−0.250 − 1.09i)12-s + (0.893 + 1.12i)13-s + (−0.0921 − 0.281i)14-s + (−0.0985 + 0.123i)15-s + (−0.671 − 0.323i)16-s + (−0.0600 − 0.263i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.833 + 0.552i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.833 + 0.552i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.87407 - 0.564156i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.87407 - 0.564156i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (17.3 + 6.46i)T \) |
good | 2 | \( 1 + (-0.521 - 0.654i)T + (-1.78 + 7.79i)T^{2} \) |
| 3 | \( 1 + (-5.78 + 2.78i)T + (16.8 - 21.1i)T^{2} \) |
| 5 | \( 1 + (1.28 - 0.620i)T + (77.9 - 97.7i)T^{2} \) |
| 11 | \( 1 + (-33.8 - 42.5i)T + (-296. + 1.29e3i)T^{2} \) |
| 13 | \( 1 + (-41.8 - 52.5i)T + (-488. + 2.14e3i)T^{2} \) |
| 17 | \( 1 + (4.21 + 18.4i)T + (-4.42e3 + 2.13e3i)T^{2} \) |
| 19 | \( 1 + 58.6T + 6.85e3T^{2} \) |
| 23 | \( 1 + (4.27 - 18.7i)T + (-1.09e4 - 5.27e3i)T^{2} \) |
| 29 | \( 1 + (50.8 + 222. i)T + (-2.19e4 + 1.05e4i)T^{2} \) |
| 31 | \( 1 - 165.T + 2.97e4T^{2} \) |
| 37 | \( 1 + (53.1 + 232. i)T + (-4.56e4 + 2.19e4i)T^{2} \) |
| 41 | \( 1 + (174. - 84.1i)T + (4.29e4 - 5.38e4i)T^{2} \) |
| 43 | \( 1 + (206. + 99.2i)T + (4.95e4 + 6.21e4i)T^{2} \) |
| 47 | \( 1 + (-153. - 192. i)T + (-2.31e4 + 1.01e5i)T^{2} \) |
| 53 | \( 1 + (-118. + 518. i)T + (-1.34e5 - 6.45e4i)T^{2} \) |
| 59 | \( 1 + (150. + 72.6i)T + (1.28e5 + 1.60e5i)T^{2} \) |
| 61 | \( 1 + (-100. - 441. i)T + (-2.04e5 + 9.84e4i)T^{2} \) |
| 67 | \( 1 - 153.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (-137. + 603. i)T + (-3.22e5 - 1.55e5i)T^{2} \) |
| 73 | \( 1 + (247. - 310. i)T + (-8.65e4 - 3.79e5i)T^{2} \) |
| 79 | \( 1 + 981.T + 4.93e5T^{2} \) |
| 83 | \( 1 + (258. - 323. i)T + (-1.27e5 - 5.57e5i)T^{2} \) |
| 89 | \( 1 + (-833. + 1.04e3i)T + (-1.56e5 - 6.87e5i)T^{2} \) |
| 97 | \( 1 - 1.06e3T + 9.12e5T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.84769578443531604780848703319, −13.90672022057247656768345711198, −13.18503549123516474997914188987, −11.57372597370255763306962354823, −9.930625738932914880452449645787, −9.024154336591564335652826358093, −7.26728510291886257183025555205, −6.36991583422893803302565835533, −4.04935265947112014302739217826, −1.87852104001168077052467156992,
3.02473460961902403442396022372, 3.78891441160254439427118816650, 6.34237739097525762374957921302, 8.307757775151763344685579900732, 8.832319896519910287591946458323, 10.42692125348362760059685211353, 11.87223962454888067422049527937, 13.06659644293220445346626529885, 13.93632411516096415053237228724, 15.31823218414518582235934189867