| L(s) = 1 | + (3.00 + 1.73i)2-s + (−5.19 − 0.0205i)3-s + (2.01 + 3.48i)4-s + (−6.93 − 1.85i)5-s + (−15.5 − 9.07i)6-s + (16.2 − 4.34i)7-s − 13.7i·8-s + (26.9 + 0.213i)9-s + (−17.6 − 17.6i)10-s + (57.9 − 15.5i)11-s + (−10.3 − 18.1i)12-s + (8.10 + 14.0i)13-s + (56.2 + 15.0i)14-s + (35.9 + 9.79i)15-s + (39.9 − 69.2i)16-s + (−8.20 + 69.6i)17-s + ⋯ |
| L(s) = 1 | + (1.06 + 0.613i)2-s + (−0.999 − 0.00396i)3-s + (0.251 + 0.435i)4-s + (−0.620 − 0.166i)5-s + (−1.05 − 0.617i)6-s + (0.875 − 0.234i)7-s − 0.608i·8-s + (0.999 + 0.00792i)9-s + (−0.556 − 0.556i)10-s + (1.58 − 0.425i)11-s + (−0.249 − 0.436i)12-s + (0.172 + 0.299i)13-s + (1.07 + 0.287i)14-s + (0.619 + 0.168i)15-s + (0.624 − 1.08i)16-s + (−0.117 + 0.993i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.159i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.987 + 0.159i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.13405 - 0.171666i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.13405 - 0.171666i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (5.19 + 0.0205i)T \) |
| 17 | \( 1 + (8.20 - 69.6i)T \) |
| good | 2 | \( 1 + (-3.00 - 1.73i)T + (4 + 6.92i)T^{2} \) |
| 5 | \( 1 + (6.93 + 1.85i)T + (108. + 62.5i)T^{2} \) |
| 7 | \( 1 + (-16.2 + 4.34i)T + (297. - 171.5i)T^{2} \) |
| 11 | \( 1 + (-57.9 + 15.5i)T + (1.15e3 - 665.5i)T^{2} \) |
| 13 | \( 1 + (-8.10 - 14.0i)T + (-1.09e3 + 1.90e3i)T^{2} \) |
| 19 | \( 1 + 89.3iT - 6.85e3T^{2} \) |
| 23 | \( 1 + (-15.6 + 58.2i)T + (-1.05e4 - 6.08e3i)T^{2} \) |
| 29 | \( 1 + (51.1 + 190. i)T + (-2.11e4 + 1.21e4i)T^{2} \) |
| 31 | \( 1 + (-209. - 56.1i)T + (2.57e4 + 1.48e4i)T^{2} \) |
| 37 | \( 1 + (-149. + 149. i)T - 5.06e4iT^{2} \) |
| 41 | \( 1 + (-11.0 + 41.3i)T + (-5.96e4 - 3.44e4i)T^{2} \) |
| 43 | \( 1 + (-148. - 85.8i)T + (3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 + (241. - 418. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 - 551. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + (361. - 208. i)T + (1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-849. + 227. i)T + (1.96e5 - 1.13e5i)T^{2} \) |
| 67 | \( 1 + (250. + 433. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + (-45.7 + 45.7i)T - 3.57e5iT^{2} \) |
| 73 | \( 1 + (-179. + 179. i)T - 3.89e5iT^{2} \) |
| 79 | \( 1 + (835. - 223. i)T + (4.26e5 - 2.46e5i)T^{2} \) |
| 83 | \( 1 + (-808. - 466. i)T + (2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 + 950.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-130. - 486. i)T + (-7.90e5 + 4.56e5i)T^{2} \) |
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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
−12.45601643472442918924781157344, −11.65958353609545167841853181945, −10.92343324078715151205378837725, −9.429695717999442441667045562812, −7.934346303400009890820314661613, −6.67289243515032088000859704009, −5.98602423999313815550957746588, −4.50633139173788960572199070480, −4.11619043965986382822296831185, −0.996948422018045515577164803888,
1.55879005747064866612268715220, 3.65128825712357965996758466610, 4.59438404458352146367527460588, 5.60065009266765638897150881908, 6.95346247791987409052884629670, 8.249785196946201760262444828625, 9.818953033565940836714909455766, 11.23443799395705090046506716290, 11.65592977539684764576509828067, 12.17903669019182818167853677267
