| L(s) = 1 | + (7.27 − 4.67i)2-s + (1.28 − 8.90i)3-s + (17.7 − 38.8i)4-s + (−64.7 + 19.0i)5-s + (−32.3 − 70.7i)6-s + (−107. − 123. i)7-s + (−13.1 − 91.6i)8-s + (−77.7 − 22.8i)9-s + (−381. + 440. i)10-s + (−320. − 205. i)11-s + (−323. − 207. i)12-s + (686. − 792. i)13-s + (−1.35e3 − 397. i)14-s + (86.4 + 601. i)15-s + (371. + 428. i)16-s + (194. + 426. i)17-s + ⋯ |
| L(s) = 1 | + (1.28 − 0.826i)2-s + (0.0821 − 0.571i)3-s + (0.554 − 1.21i)4-s + (−1.15 + 0.340i)5-s + (−0.366 − 0.802i)6-s + (−0.825 − 0.952i)7-s + (−0.0727 − 0.506i)8-s + (−0.319 − 0.0939i)9-s + (−1.20 + 1.39i)10-s + (−0.798 − 0.512i)11-s + (−0.648 − 0.416i)12-s + (1.12 − 1.29i)13-s + (−1.84 − 0.542i)14-s + (0.0991 + 0.689i)15-s + (0.362 + 0.418i)16-s + (0.163 + 0.357i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 + 0.114i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.993 + 0.114i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.123408 - 2.14163i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.123408 - 2.14163i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.28 + 8.90i)T \) |
| 23 | \( 1 + (-2.51e3 - 328. i)T \) |
| good | 2 | \( 1 + (-7.27 + 4.67i)T + (13.2 - 29.1i)T^{2} \) |
| 5 | \( 1 + (64.7 - 19.0i)T + (2.62e3 - 1.68e3i)T^{2} \) |
| 7 | \( 1 + (107. + 123. i)T + (-2.39e3 + 1.66e4i)T^{2} \) |
| 11 | \( 1 + (320. + 205. i)T + (6.69e4 + 1.46e5i)T^{2} \) |
| 13 | \( 1 + (-686. + 792. i)T + (-5.28e4 - 3.67e5i)T^{2} \) |
| 17 | \( 1 + (-194. - 426. i)T + (-9.29e5 + 1.07e6i)T^{2} \) |
| 19 | \( 1 + (-547. + 1.19e3i)T + (-1.62e6 - 1.87e6i)T^{2} \) |
| 29 | \( 1 + (1.52e3 + 3.33e3i)T + (-1.34e7 + 1.55e7i)T^{2} \) |
| 31 | \( 1 + (-719. - 5.00e3i)T + (-2.74e7 + 8.06e6i)T^{2} \) |
| 37 | \( 1 + (1.29e4 + 3.81e3i)T + (5.83e7 + 3.74e7i)T^{2} \) |
| 41 | \( 1 + (-1.07e4 + 3.15e3i)T + (9.74e7 - 6.26e7i)T^{2} \) |
| 43 | \( 1 + (-2.44e3 + 1.69e4i)T + (-1.41e8 - 4.14e7i)T^{2} \) |
| 47 | \( 1 + 1.71e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + (-1.63e4 - 1.88e4i)T + (-5.95e7 + 4.13e8i)T^{2} \) |
| 59 | \( 1 + (-9.66e3 + 1.11e4i)T + (-1.01e8 - 7.07e8i)T^{2} \) |
| 61 | \( 1 + (-1.19e3 - 8.33e3i)T + (-8.10e8 + 2.37e8i)T^{2} \) |
| 67 | \( 1 + (-3.38e4 + 2.17e4i)T + (5.60e8 - 1.22e9i)T^{2} \) |
| 71 | \( 1 + (-1.51e4 + 9.74e3i)T + (7.49e8 - 1.64e9i)T^{2} \) |
| 73 | \( 1 + (1.64e4 - 3.61e4i)T + (-1.35e9 - 1.56e9i)T^{2} \) |
| 79 | \( 1 + (-3.68e4 + 4.25e4i)T + (-4.37e8 - 3.04e9i)T^{2} \) |
| 83 | \( 1 + (6.48e4 + 1.90e4i)T + (3.31e9 + 2.12e9i)T^{2} \) |
| 89 | \( 1 + (9.63e3 - 6.69e4i)T + (-5.35e9 - 1.57e9i)T^{2} \) |
| 97 | \( 1 + (1.08e5 - 3.18e4i)T + (7.22e9 - 4.64e9i)T^{2} \) |
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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
−13.21246766944159704974853065962, −12.37512663289366259951038434719, −11.09102083899722986074139900022, −10.57841660350474897139249159365, −8.273355771523247223961217669257, −7.06973947781723846730626756994, −5.53049795069020110882544759329, −3.72847048634862688255090591833, −3.07786108773182464444818612278, −0.63890005731369136455901924826,
3.25422064605757243547315199509, 4.34258583567363835981444072496, 5.53565423604723337950961374652, 6.85185721380093203605165769222, 8.249938478700829197356887622850, 9.548434525164134569779163543762, 11.39458277511758882097966137544, 12.34998513859028046398966077319, 13.23803062665896824553694702313, 14.52181662378086143054954500485
