L(s) = 1 | + (1.5 + 2.59i)3-s + (10.6 − 18.5i)5-s + (18.2 + 3.19i)7-s + (−4.5 + 7.79i)9-s + (5.35 + 9.26i)11-s + 8.31·13-s + 64.1·15-s + (−0.742 − 1.28i)17-s + (80.7 − 139. i)19-s + (19.0 + 52.1i)21-s + (−61.9 + 107. i)23-s + (−166. − 288. i)25-s − 27·27-s − 222.·29-s + (101. + 176. i)31-s + ⋯ |
L(s) = 1 | + (0.288 + 0.499i)3-s + (0.956 − 1.65i)5-s + (0.984 + 0.172i)7-s + (−0.166 + 0.288i)9-s + (0.146 + 0.254i)11-s + 0.177·13-s + 1.10·15-s + (−0.0105 − 0.0183i)17-s + (0.975 − 1.68i)19-s + (0.198 + 0.542i)21-s + (−0.561 + 0.973i)23-s + (−1.33 − 2.30i)25-s − 0.192·27-s − 1.42·29-s + (0.590 + 1.02i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.805 + 0.592i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.805 + 0.592i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.743865452\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.743865452\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.5 - 2.59i)T \) |
| 7 | \( 1 + (-18.2 - 3.19i)T \) |
good | 5 | \( 1 + (-10.6 + 18.5i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (-5.35 - 9.26i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 - 8.31T + 2.19e3T^{2} \) |
| 17 | \( 1 + (0.742 + 1.28i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-80.7 + 139. i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (61.9 - 107. i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + 222.T + 2.43e4T^{2} \) |
| 31 | \( 1 + (-101. - 176. i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-170. + 295. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 - 147.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 51.3T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-134. + 233. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-216. - 374. i)T + (-7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-15.3 - 26.6i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-296. + 513. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-207. - 358. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + 294.T + 3.57e5T^{2} \) |
| 73 | \( 1 + (-297. - 514. i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (415. - 720. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + 326.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (-630. + 1.09e3i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + 1.05e3T + 9.12e5T^{2} \) |
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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
−11.07783766040937718640314741264, −9.763147593807423001580504615618, −9.177879476577179061091301940139, −8.522937604884288189463361793219, −7.41901828105466205218190100505, −5.64638934689864623197982019901, −5.10796200301954845500892144145, −4.15990444283743466823064403846, −2.23755259585099508961440100353, −1.06659293848363275262715768338,
1.55645030340641848042154032163, 2.58732358850318283383138095580, 3.82171439316299136748098236939, 5.65442511572058073763705673474, 6.34667596239950781653967157028, 7.44827714318098950556414238758, 8.121909469567608363274176720481, 9.562222329040325283633462328100, 10.31170885796843161147328885469, 11.18420832575973208106880573114