| L(s) = 1 | + (−3.77 − 1.82i)2-s + (−2.29 + 2.87i)3-s + (5.98 + 7.50i)4-s + (15.1 + 7.31i)5-s + (13.9 − 6.70i)6-s + (−0.697 + 0.874i)7-s + (−1.49 − 6.53i)8-s + (2.99 + 13.1i)9-s + (−44.0 − 55.2i)10-s + (−11.5 + 50.6i)11-s − 35.3·12-s + (12.0 − 52.9i)13-s + (4.22 − 2.03i)14-s + (−55.8 + 26.9i)15-s + (10.8 − 47.4i)16-s − 17.6·17-s + ⋯ |
| L(s) = 1 | + (−1.33 − 0.643i)2-s + (−0.441 + 0.553i)3-s + (0.748 + 0.938i)4-s + (1.35 + 0.653i)5-s + (0.946 − 0.455i)6-s + (−0.0376 + 0.0472i)7-s + (−0.0659 − 0.288i)8-s + (0.110 + 0.485i)9-s + (−1.39 − 1.74i)10-s + (−0.316 + 1.38i)11-s − 0.849·12-s + (0.257 − 1.12i)13-s + (0.0806 − 0.0388i)14-s + (−0.961 + 0.463i)15-s + (0.169 − 0.741i)16-s − 0.252·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.782 - 0.622i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.782 - 0.622i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.612332 + 0.213865i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.612332 + 0.213865i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 29 | \( 1 + (-154. + 19.8i)T \) |
| good | 2 | \( 1 + (3.77 + 1.82i)T + (4.98 + 6.25i)T^{2} \) |
| 3 | \( 1 + (2.29 - 2.87i)T + (-6.00 - 26.3i)T^{2} \) |
| 5 | \( 1 + (-15.1 - 7.31i)T + (77.9 + 97.7i)T^{2} \) |
| 7 | \( 1 + (0.697 - 0.874i)T + (-76.3 - 334. i)T^{2} \) |
| 11 | \( 1 + (11.5 - 50.6i)T + (-1.19e3 - 577. i)T^{2} \) |
| 13 | \( 1 + (-12.0 + 52.9i)T + (-1.97e3 - 953. i)T^{2} \) |
| 17 | \( 1 + 17.6T + 4.91e3T^{2} \) |
| 19 | \( 1 + (-52.4 - 65.7i)T + (-1.52e3 + 6.68e3i)T^{2} \) |
| 23 | \( 1 + (-31.8 + 15.3i)T + (7.58e3 - 9.51e3i)T^{2} \) |
| 31 | \( 1 + (292. + 141. i)T + (1.85e4 + 2.32e4i)T^{2} \) |
| 37 | \( 1 + (38.8 + 170. i)T + (-4.56e4 + 2.19e4i)T^{2} \) |
| 41 | \( 1 - 318.T + 6.89e4T^{2} \) |
| 43 | \( 1 + (-258. + 124. i)T + (4.95e4 - 6.21e4i)T^{2} \) |
| 47 | \( 1 + (-40.5 + 177. i)T + (-9.35e4 - 4.50e4i)T^{2} \) |
| 53 | \( 1 + (445. + 214. i)T + (9.28e4 + 1.16e5i)T^{2} \) |
| 59 | \( 1 - 225.T + 2.05e5T^{2} \) |
| 61 | \( 1 + (-485. + 608. i)T + (-5.05e4 - 2.21e5i)T^{2} \) |
| 67 | \( 1 + (-67.6 - 296. i)T + (-2.70e5 + 1.30e5i)T^{2} \) |
| 71 | \( 1 + (46.8 - 205. i)T + (-3.22e5 - 1.55e5i)T^{2} \) |
| 73 | \( 1 + (-316. + 152. i)T + (2.42e5 - 3.04e5i)T^{2} \) |
| 79 | \( 1 + (109. + 478. i)T + (-4.44e5 + 2.13e5i)T^{2} \) |
| 83 | \( 1 + (334. + 419. i)T + (-1.27e5 + 5.57e5i)T^{2} \) |
| 89 | \( 1 + (838. + 403. i)T + (4.39e5 + 5.51e5i)T^{2} \) |
| 97 | \( 1 + (-916. - 1.14e3i)T + (-2.03e5 + 8.89e5i)T^{2} \) |
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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
−17.35846829791676073877120087168, −15.95243638042346085223365221260, −14.38997117741917831645413060022, −12.77852676669354296072704310280, −10.93824409661060745025266084736, −10.22278838953532914855363554272, −9.504149616874802254861052459593, −7.58603904557574178489372615723, −5.49377676475354648386498716993, −2.20391352777962249055404028001,
1.11464208631827395116616453381, 5.81180662487762204203053072624, 6.88839194959752635674090948047, 8.761633160334978353913257805910, 9.487996621055855715888855891657, 11.08655032174895221084500981856, 12.88486014856013974190304162878, 13.97235772709958688660296623837, 15.98506434094544127479961209307, 16.76904967020112585655860530265
