| 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
−15.31823218414518582235934189867, −13.93632411516096415053237228724, −13.06659644293220445346626529885, −11.87223962454888067422049527937, −10.42692125348362760059685211353, −8.832319896519910287591946458323, −8.307757775151763344685579900732, −6.34237739097525762374957921302, −3.78891441160254439427118816650, −3.02473460961902403442396022372,
1.87852104001168077052467156992, 4.04935265947112014302739217826, 6.36991583422893803302565835533, 7.26728510291886257183025555205, 9.024154336591564335652826358093, 9.930625738932914880452449645787, 11.57372597370255763306962354823, 13.18503549123516474997914188987, 13.90672022057247656768345711198, 14.84769578443531604780848703319
