| L(s) = 1 | + (1.72 + 1.01i)2-s + (1.92 + 3.50i)4-s + (−2.16 + 6.65i)7-s + (−0.247 + 7.99i)8-s + (7.28 − 5.29i)9-s + (−3.79 + 10.3i)11-s + (−10.5 + 9.26i)14-s + (−8.56 + 13.5i)16-s + (17.9 − 1.69i)18-s + (−17.0 + 13.9i)22-s + 4.03i·23-s + (7.72 + 23.7i)25-s + (−27.5 + 5.25i)28-s + (13.9 + 4.51i)29-s + (−28.5 + 14.5i)32-s + ⋯ |
| L(s) = 1 | + (0.860 + 0.508i)2-s + (0.482 + 0.876i)4-s + (−0.309 + 0.951i)7-s + (−0.0309 + 0.999i)8-s + (0.809 − 0.587i)9-s + (−0.344 + 0.938i)11-s + (−0.750 + 0.661i)14-s + (−0.535 + 0.844i)16-s + (0.995 − 0.0942i)18-s + (−0.774 + 0.632i)22-s + 0.175i·23-s + (0.309 + 0.951i)25-s + (−0.982 + 0.187i)28-s + (0.479 + 0.155i)29-s + (−0.890 + 0.454i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.218 - 0.975i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.218 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.61081 + 2.01170i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.61081 + 2.01170i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.72 - 1.01i)T \) |
| 7 | \( 1 + (2.16 - 6.65i)T \) |
| 11 | \( 1 + (3.79 - 10.3i)T \) |
| good | 3 | \( 1 + (-7.28 + 5.29i)T^{2} \) |
| 5 | \( 1 + (-7.72 - 23.7i)T^{2} \) |
| 13 | \( 1 + (52.2 - 160. i)T^{2} \) |
| 17 | \( 1 + (89.3 + 274. i)T^{2} \) |
| 19 | \( 1 + (292. - 212. i)T^{2} \) |
| 23 | \( 1 - 4.03iT - 529T^{2} \) |
| 29 | \( 1 + (-13.9 - 4.51i)T + (680. + 494. i)T^{2} \) |
| 31 | \( 1 + (296. - 913. i)T^{2} \) |
| 37 | \( 1 + (-22.2 + 68.6i)T + (-1.10e3 - 804. i)T^{2} \) |
| 41 | \( 1 + (-1.35e3 + 988. i)T^{2} \) |
| 43 | \( 1 - 9.59T + 1.84e3T^{2} \) |
| 47 | \( 1 + (-1.78e3 + 1.29e3i)T^{2} \) |
| 53 | \( 1 + (-54.2 + 39.4i)T + (868. - 2.67e3i)T^{2} \) |
| 59 | \( 1 + (-2.81e3 - 2.04e3i)T^{2} \) |
| 61 | \( 1 + (1.14e3 + 3.53e3i)T^{2} \) |
| 67 | \( 1 + 17.9iT - 4.48e3T^{2} \) |
| 71 | \( 1 + (-79.6 + 109. i)T + (-1.55e3 - 4.79e3i)T^{2} \) |
| 73 | \( 1 + (-4.31e3 - 3.13e3i)T^{2} \) |
| 79 | \( 1 + (-1.13 + 0.823i)T + (1.92e3 - 5.93e3i)T^{2} \) |
| 83 | \( 1 + (-2.12e3 - 6.55e3i)T^{2} \) |
| 89 | \( 1 - 7.92e3T^{2} \) |
| 97 | \( 1 + (-2.90e3 + 8.94e3i)T^{2} \) |
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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
−12.18420055197837156057082821914, −11.04518002947137632267974206172, −9.753091485429978623287832863767, −8.871585311217378268224584676442, −7.59752578190093283129160058345, −6.80858462379032023695867926737, −5.75576683783093954822408554424, −4.74918458439529716216669103494, −3.55265313850399882143580943897, −2.18255204209033464938752976686,
1.00229200656210399053777468339, 2.72509536766783651161816015215, 3.98473790559794955034361093654, 4.88357602304156357927704856669, 6.18759787419397384278873016440, 7.09586502970071414671152088925, 8.252921057642354797701397236849, 9.821185728476640948568509013155, 10.43170945348004314564972838119, 11.15907508870110536347323217068
