| L(s) = 1 | + (−4.85 − 1.30i)2-s + (−11.3 − 6.52i)3-s + (8.02 + 4.63i)4-s + (−23.1 + 6.21i)5-s + (46.4 + 46.4i)6-s + (29.0 − 50.3i)7-s + (23.9 + 23.9i)8-s + (44.7 + 77.5i)9-s + 120.·10-s + 128. i·11-s + (−60.5 − 104. i)12-s + (−46.1 + 12.3i)13-s + (−206. + 206. i)14-s + (302. + 81.1i)15-s + (−159. − 275. i)16-s + (68.3 − 255. i)17-s + ⋯ |
| L(s) = 1 | + (−1.21 − 0.325i)2-s + (−1.25 − 0.725i)3-s + (0.501 + 0.289i)4-s + (−0.927 + 0.248i)5-s + (1.28 + 1.28i)6-s + (0.593 − 1.02i)7-s + (0.374 + 0.374i)8-s + (0.552 + 0.957i)9-s + 1.20·10-s + 1.05i·11-s + (−0.420 − 0.727i)12-s + (−0.272 + 0.0731i)13-s + (−1.05 + 1.05i)14-s + (1.34 + 0.360i)15-s + (−0.621 − 1.07i)16-s + (0.236 − 0.882i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.687 - 0.726i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.687 - 0.726i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.197609 + 0.0850529i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.197609 + 0.0850529i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 + (-1.24e3 - 572. i)T \) |
| good | 2 | \( 1 + (4.85 + 1.30i)T + (13.8 + 8i)T^{2} \) |
| 3 | \( 1 + (11.3 + 6.52i)T + (40.5 + 70.1i)T^{2} \) |
| 5 | \( 1 + (23.1 - 6.21i)T + (541. - 312.5i)T^{2} \) |
| 7 | \( 1 + (-29.0 + 50.3i)T + (-1.20e3 - 2.07e3i)T^{2} \) |
| 11 | \( 1 - 128. iT - 1.46e4T^{2} \) |
| 13 | \( 1 + (46.1 - 12.3i)T + (2.47e4 - 1.42e4i)T^{2} \) |
| 17 | \( 1 + (-68.3 + 255. i)T + (-7.23e4 - 4.17e4i)T^{2} \) |
| 19 | \( 1 + (259. - 69.3i)T + (1.12e5 - 6.51e4i)T^{2} \) |
| 23 | \( 1 + (-706. - 706. i)T + 2.79e5iT^{2} \) |
| 29 | \( 1 + (638. - 638. i)T - 7.07e5iT^{2} \) |
| 31 | \( 1 + (-467. + 467. i)T - 9.23e5iT^{2} \) |
| 41 | \( 1 + (-1.22e3 - 705. i)T + (1.41e6 + 2.44e6i)T^{2} \) |
| 43 | \( 1 + (-1.41e3 - 1.41e3i)T + 3.41e6iT^{2} \) |
| 47 | \( 1 + 3.96e3T + 4.87e6T^{2} \) |
| 53 | \( 1 + (2.47e3 + 4.29e3i)T + (-3.94e6 + 6.83e6i)T^{2} \) |
| 59 | \( 1 + (560. - 2.09e3i)T + (-1.04e7 - 6.05e6i)T^{2} \) |
| 61 | \( 1 + (-130. - 485. i)T + (-1.19e7 + 6.92e6i)T^{2} \) |
| 67 | \( 1 + (-135. - 78.0i)T + (1.00e7 + 1.74e7i)T^{2} \) |
| 71 | \( 1 + (4.82e3 - 8.35e3i)T + (-1.27e7 - 2.20e7i)T^{2} \) |
| 73 | \( 1 + 1.64e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 + (4.38e3 - 1.17e3i)T + (3.37e7 - 1.94e7i)T^{2} \) |
| 83 | \( 1 + (-3.83e3 - 6.64e3i)T + (-2.37e7 + 4.11e7i)T^{2} \) |
| 89 | \( 1 + (4.79e3 + 1.28e3i)T + (5.43e7 + 3.13e7i)T^{2} \) |
| 97 | \( 1 + (-5.03e3 - 5.03e3i)T + 8.85e7iT^{2} \) |
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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
−16.42405895547287813830813457195, −14.72769356710134027172059644963, −13.00280624002719583893628798770, −11.48935025942651913871583817064, −11.14537079122152775891007878234, −9.731792615939413951590525331622, −7.70728180808483221225025696493, −7.14698716991927426711060538089, −4.78958402734948954800113788479, −1.20018583546295580266138849765,
0.30796891178015087727037213330, 4.49578530644752215456534119351, 6.09275265391060888074277085690, 8.040393068457900625562872173670, 8.980950423506214349672748261594, 10.60991777137612647208536356503, 11.35643872340898994569560305665, 12.58595047194520493766986590957, 15.00786514299142908162120775079, 15.92010300789454355159382612978
