| L(s) = 1 | + (−0.0253 + 0.127i)3-s + (−2.11 + 1.41i)5-s + (5.83 + 3.90i)7-s + (8.29 + 3.43i)9-s + (−6.06 + 1.20i)11-s + (15.1 + 15.1i)13-s + (−0.126 − 0.305i)15-s + (14.8 − 8.30i)17-s + (−15.6 + 6.47i)19-s + (−0.644 + 0.644i)21-s + (−4.90 − 24.6i)23-s + (−7.08 + 17.0i)25-s + (−1.29 + 1.94i)27-s + (3.88 + 5.81i)29-s + (11.8 + 2.36i)31-s + ⋯ |
| L(s) = 1 | + (−0.00844 + 0.0424i)3-s + (−0.423 + 0.283i)5-s + (0.833 + 0.557i)7-s + (0.922 + 0.381i)9-s + (−0.551 + 0.109i)11-s + (1.16 + 1.16i)13-s + (−0.00844 − 0.0203i)15-s + (0.872 − 0.488i)17-s + (−0.822 + 0.340i)19-s + (−0.0306 + 0.0306i)21-s + (−0.213 − 1.07i)23-s + (−0.283 + 0.683i)25-s + (−0.0480 + 0.0718i)27-s + (0.133 + 0.200i)29-s + (0.383 + 0.0762i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.722 - 0.691i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.722 - 0.691i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.36854 + 0.549075i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.36854 + 0.549075i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 + (-14.8 + 8.30i)T \) |
| good | 3 | \( 1 + (0.0253 - 0.127i)T + (-8.31 - 3.44i)T^{2} \) |
| 5 | \( 1 + (2.11 - 1.41i)T + (9.56 - 23.0i)T^{2} \) |
| 7 | \( 1 + (-5.83 - 3.90i)T + (18.7 + 45.2i)T^{2} \) |
| 11 | \( 1 + (6.06 - 1.20i)T + (111. - 46.3i)T^{2} \) |
| 13 | \( 1 + (-15.1 - 15.1i)T + 169iT^{2} \) |
| 19 | \( 1 + (15.6 - 6.47i)T + (255. - 255. i)T^{2} \) |
| 23 | \( 1 + (4.90 + 24.6i)T + (-488. + 202. i)T^{2} \) |
| 29 | \( 1 + (-3.88 - 5.81i)T + (-321. + 776. i)T^{2} \) |
| 31 | \( 1 + (-11.8 - 2.36i)T + (887. + 367. i)T^{2} \) |
| 37 | \( 1 + (-9.88 + 49.7i)T + (-1.26e3 - 523. i)T^{2} \) |
| 41 | \( 1 + (13.2 + 8.86i)T + (643. + 1.55e3i)T^{2} \) |
| 43 | \( 1 + (60.9 + 25.2i)T + (1.30e3 + 1.30e3i)T^{2} \) |
| 47 | \( 1 + (19.8 + 19.8i)T + 2.20e3iT^{2} \) |
| 53 | \( 1 + (-3.12 + 1.29i)T + (1.98e3 - 1.98e3i)T^{2} \) |
| 59 | \( 1 + (-24.9 + 60.1i)T + (-2.46e3 - 2.46e3i)T^{2} \) |
| 61 | \( 1 + (-0.934 + 1.39i)T + (-1.42e3 - 3.43e3i)T^{2} \) |
| 67 | \( 1 - 107. iT - 4.48e3T^{2} \) |
| 71 | \( 1 + (-8.16 + 41.0i)T + (-4.65e3 - 1.92e3i)T^{2} \) |
| 73 | \( 1 + (19.3 - 12.9i)T + (2.03e3 - 4.92e3i)T^{2} \) |
| 79 | \( 1 + (-147. + 29.3i)T + (5.76e3 - 2.38e3i)T^{2} \) |
| 83 | \( 1 + (-13.1 - 31.8i)T + (-4.87e3 + 4.87e3i)T^{2} \) |
| 89 | \( 1 + (-23.1 + 23.1i)T - 7.92e3iT^{2} \) |
| 97 | \( 1 + (50.3 + 75.4i)T + (-3.60e3 + 8.69e3i)T^{2} \) |
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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
−13.08010241645661096184428285483, −11.98282943947529734194298889369, −11.11158062478946884326207284961, −10.15731114839890564228135454169, −8.762265493393549273653887715341, −7.84024335826964428094849855764, −6.65622574934475864982685717540, −5.14135535343477442837953687987, −3.91008706397844117751002227762, −1.90683022474659280237489496343,
1.19104149262432096544191086141, 3.58191445505215681699515137692, 4.80087155467356690277423493917, 6.25663532169898557016789508983, 7.80423433643530130795715397208, 8.272475414337249255797251854096, 9.960229517556065928526023478242, 10.75688160558822036801037304939, 11.84704751495346632706260523836, 12.95121573421839875049485466974
