| L(s) = 1 | + (14.3 − 2.52i)2-s + (157. + 188. i)3-s + (−763. + 277. i)4-s + (1.31e3 + 477. i)5-s + (2.73e3 + 2.29e3i)6-s + (675. − 1.17e3i)7-s + (−2.31e4 + 1.33e4i)8-s + (−223. + 1.26e3i)9-s + (2.00e4 + 3.52e3i)10-s + (−3.80e4 − 6.58e4i)11-s + (−1.72e5 − 9.98e4i)12-s + (3.04e5 − 3.62e5i)13-s + (6.71e3 − 1.84e4i)14-s + (1.17e5 + 3.22e5i)15-s + (3.40e5 − 2.85e5i)16-s + (−2.00e5 − 1.13e6i)17-s + ⋯ |
| L(s) = 1 | + (0.447 − 0.0788i)2-s + (0.649 + 0.774i)3-s + (−0.745 + 0.271i)4-s + (0.420 + 0.152i)5-s + (0.351 + 0.295i)6-s + (0.0402 − 0.0696i)7-s + (−0.705 + 0.407i)8-s + (−0.00379 + 0.0214i)9-s + (0.200 + 0.0352i)10-s + (−0.236 − 0.408i)11-s + (−0.694 − 0.401i)12-s + (0.820 − 0.977i)13-s + (0.0124 − 0.0343i)14-s + (0.154 + 0.424i)15-s + (0.324 − 0.272i)16-s + (−0.141 − 0.802i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0776i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.996 - 0.0776i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(3.01748 + 0.117381i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.01748 + 0.117381i\) |
| \(L(6)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-1.31e3 - 477. i)T \) |
| 19 | \( 1 + (-2.20e6 - 1.11e6i)T \) |
| good | 2 | \( 1 + (-14.3 + 2.52i)T + (962. - 350. i)T^{2} \) |
| 3 | \( 1 + (-157. - 188. i)T + (-1.02e4 + 5.81e4i)T^{2} \) |
| 7 | \( 1 + (-675. + 1.17e3i)T + (-1.41e8 - 2.44e8i)T^{2} \) |
| 11 | \( 1 + (3.80e4 + 6.58e4i)T + (-1.29e10 + 2.24e10i)T^{2} \) |
| 13 | \( 1 + (-3.04e5 + 3.62e5i)T + (-2.39e10 - 1.35e11i)T^{2} \) |
| 17 | \( 1 + (2.00e5 + 1.13e6i)T + (-1.89e12 + 6.89e11i)T^{2} \) |
| 23 | \( 1 + (-3.15e6 + 1.14e6i)T + (3.17e13 - 2.66e13i)T^{2} \) |
| 29 | \( 1 + (2.84e7 + 5.01e6i)T + (3.95e14 + 1.43e14i)T^{2} \) |
| 31 | \( 1 + (-1.84e7 - 1.06e7i)T + (4.09e14 + 7.09e14i)T^{2} \) |
| 37 | \( 1 - 5.45e7iT - 4.80e15T^{2} \) |
| 41 | \( 1 + (-2.69e7 - 3.20e7i)T + (-2.33e15 + 1.32e16i)T^{2} \) |
| 43 | \( 1 + (-1.06e8 - 3.88e7i)T + (1.65e16 + 1.38e16i)T^{2} \) |
| 47 | \( 1 + (-4.55e7 + 2.58e8i)T + (-4.94e16 - 1.79e16i)T^{2} \) |
| 53 | \( 1 + (-4.50e7 - 1.23e8i)T + (-1.33e17 + 1.12e17i)T^{2} \) |
| 59 | \( 1 + (4.62e8 - 8.15e7i)T + (4.80e17 - 1.74e17i)T^{2} \) |
| 61 | \( 1 + (1.06e9 - 3.87e8i)T + (5.46e17 - 4.58e17i)T^{2} \) |
| 67 | \( 1 + (-4.23e8 - 7.46e7i)T + (1.71e18 + 6.23e17i)T^{2} \) |
| 71 | \( 1 + (-8.39e8 + 2.30e9i)T + (-2.49e18 - 2.09e18i)T^{2} \) |
| 73 | \( 1 + (-1.84e9 + 1.54e9i)T + (7.46e17 - 4.23e18i)T^{2} \) |
| 79 | \( 1 + (-3.56e9 - 4.24e9i)T + (-1.64e18 + 9.32e18i)T^{2} \) |
| 83 | \( 1 + (-1.38e9 + 2.39e9i)T + (-7.75e18 - 1.34e19i)T^{2} \) |
| 89 | \( 1 + (-5.72e9 + 6.82e9i)T + (-5.41e18 - 3.07e19i)T^{2} \) |
| 97 | \( 1 + (-1.54e9 + 2.71e8i)T + (6.92e19 - 2.52e19i)T^{2} \) |
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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
−12.15383530639127335366214836339, −10.77036452571322266062112670588, −9.639658763779733335023705992086, −8.907943385710312935485100936347, −7.81759170808424023045728825317, −5.94802384574253701105607592830, −4.86221284441484622021010952227, −3.58074406630352017897647684748, −2.90686965615574923798565504892, −0.75186620896623940386314606454,
1.05334531855466159563709409430, 2.15982560444578309547731796237, 3.72681008437388635793522940713, 5.02226774054130305304972348503, 6.22052144577832253978471435219, 7.50504995447322684008184809020, 8.757946238760697953515188401021, 9.470117317715289323230804719600, 10.92210718156596197886488345298, 12.43673274581648766235853870703
