| L(s) = 1 | + (0.477 + 2.70i)2-s + (−4.62 + 3.87i)3-s + (0.412 − 0.150i)4-s + (5.11 + 1.86i)5-s + (−12.7 − 10.6i)6-s + (11.7 − 20.3i)7-s + (11.6 + 20.0i)8-s + (1.63 − 9.27i)9-s + (−2.59 + 14.7i)10-s + (−23.6 − 40.8i)11-s + (−1.32 + 2.29i)12-s + (−14.0 − 11.7i)13-s + (60.6 + 22.0i)14-s + (−30.8 + 11.2i)15-s + (−46.1 + 38.7i)16-s + (16.8 + 95.7i)17-s + ⋯ |
| L(s) = 1 | + (0.168 + 0.957i)2-s + (−0.889 + 0.746i)3-s + (0.0516 − 0.0187i)4-s + (0.457 + 0.166i)5-s + (−0.864 − 0.725i)6-s + (0.633 − 1.09i)7-s + (0.512 + 0.888i)8-s + (0.0605 − 0.343i)9-s + (−0.0821 + 0.465i)10-s + (−0.646 − 1.12i)11-s + (−0.0318 + 0.0552i)12-s + (−0.299 − 0.251i)13-s + (1.15 + 0.421i)14-s + (−0.530 + 0.193i)15-s + (−0.721 + 0.605i)16-s + (0.240 + 1.36i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0208 - 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0208 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.757607 + 0.741947i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.757607 + 0.741947i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 + (25.4 + 78.8i)T \) |
| good | 2 | \( 1 + (-0.477 - 2.70i)T + (-7.51 + 2.73i)T^{2} \) |
| 3 | \( 1 + (4.62 - 3.87i)T + (4.68 - 26.5i)T^{2} \) |
| 5 | \( 1 + (-5.11 - 1.86i)T + (95.7 + 80.3i)T^{2} \) |
| 7 | \( 1 + (-11.7 + 20.3i)T + (-171.5 - 297. i)T^{2} \) |
| 11 | \( 1 + (23.6 + 40.8i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + (14.0 + 11.7i)T + (381. + 2.16e3i)T^{2} \) |
| 17 | \( 1 + (-16.8 - 95.7i)T + (-4.61e3 + 1.68e3i)T^{2} \) |
| 23 | \( 1 + (-84.7 + 30.8i)T + (9.32e3 - 7.82e3i)T^{2} \) |
| 29 | \( 1 + (-16.0 + 91.2i)T + (-2.29e4 - 8.34e3i)T^{2} \) |
| 31 | \( 1 + (142. - 246. i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 - 232.T + 5.06e4T^{2} \) |
| 41 | \( 1 + (197. - 165. i)T + (1.19e4 - 6.78e4i)T^{2} \) |
| 43 | \( 1 + (75.7 + 27.5i)T + (6.09e4 + 5.11e4i)T^{2} \) |
| 47 | \( 1 + (85.6 - 485. i)T + (-9.75e4 - 3.55e4i)T^{2} \) |
| 53 | \( 1 + (106. - 38.8i)T + (1.14e5 - 9.56e4i)T^{2} \) |
| 59 | \( 1 + (31.3 + 177. i)T + (-1.92e5 + 7.02e4i)T^{2} \) |
| 61 | \( 1 + (-259. + 94.4i)T + (1.73e5 - 1.45e5i)T^{2} \) |
| 67 | \( 1 + (-26.9 + 152. i)T + (-2.82e5 - 1.02e5i)T^{2} \) |
| 71 | \( 1 + (577. + 210. i)T + (2.74e5 + 2.30e5i)T^{2} \) |
| 73 | \( 1 + (-368. + 309. i)T + (6.75e4 - 3.83e5i)T^{2} \) |
| 79 | \( 1 + (-975. + 818. i)T + (8.56e4 - 4.85e5i)T^{2} \) |
| 83 | \( 1 + (154. - 266. i)T + (-2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 + (-32.1 - 26.9i)T + (1.22e5 + 6.94e5i)T^{2} \) |
| 97 | \( 1 + (-98.6 - 559. i)T + (-8.57e5 + 3.12e5i)T^{2} \) |
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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
−17.58571049231364178993688467743, −16.93356885182216919351569912414, −16.00046767615121531612805351352, −14.71824807933223314861108169808, −13.49479868271364836375908381551, −11.05117490820681864201941273315, −10.53308043804299441271528904101, −7.979217887726098386679367197819, −6.21168813744426679540195056024, −4.88527830381638644947394285782,
2.01401151355066298006743062483, 5.36224154073208569630255776232, 7.25709112743548963070043192620, 9.681027570734621469993570719502, 11.40181237116285070934823569994, 12.14654856562785850390565959699, 13.09514191634871887156374536847, 15.09223747983421154214837949996, 16.75368433483603873772482623653, 18.10429610197730198780188199831
