| L(s) = 1 | + (3.23 − 2.35i)2-s + (8.20 − 25.2i)3-s + (4.94 − 15.2i)4-s + (−55.6 − 5.47i)5-s + (−32.8 − 101. i)6-s + 129.·7-s + (−19.7 − 60.8i)8-s + (−373. − 271. i)9-s + (−192. + 113. i)10-s + (−270. + 196. i)11-s + (−343. − 249. i)12-s + (890. + 646. i)13-s + (419. − 304. i)14-s + (−594. + 1.35e3i)15-s + (−207. − 150. i)16-s + (−618. − 1.90e3i)17-s + ⋯ |
| L(s) = 1 | + (0.572 − 0.415i)2-s + (0.526 − 1.62i)3-s + (0.154 − 0.475i)4-s + (−0.995 − 0.0979i)5-s + (−0.372 − 1.14i)6-s + 1.00·7-s + (−0.109 − 0.336i)8-s + (−1.53 − 1.11i)9-s + (−0.610 + 0.357i)10-s + (−0.674 + 0.490i)11-s + (−0.689 − 0.500i)12-s + (1.46 + 1.06i)13-s + (0.572 − 0.415i)14-s + (−0.682 + 1.56i)15-s + (−0.202 − 0.146i)16-s + (−0.519 − 1.59i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.848 + 0.529i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.848 + 0.529i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.636030 - 2.22149i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.636030 - 2.22149i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-3.23 + 2.35i)T \) |
| 5 | \( 1 + (55.6 + 5.47i)T \) |
| good | 3 | \( 1 + (-8.20 + 25.2i)T + (-196. - 142. i)T^{2} \) |
| 7 | \( 1 - 129.T + 1.68e4T^{2} \) |
| 11 | \( 1 + (270. - 196. i)T + (4.97e4 - 1.53e5i)T^{2} \) |
| 13 | \( 1 + (-890. - 646. i)T + (1.14e5 + 3.53e5i)T^{2} \) |
| 17 | \( 1 + (618. + 1.90e3i)T + (-1.14e6 + 8.34e5i)T^{2} \) |
| 19 | \( 1 + (265. + 816. i)T + (-2.00e6 + 1.45e6i)T^{2} \) |
| 23 | \( 1 + (-1.04e3 + 756. i)T + (1.98e6 - 6.12e6i)T^{2} \) |
| 29 | \( 1 + (-1.50e3 + 4.63e3i)T + (-1.65e7 - 1.20e7i)T^{2} \) |
| 31 | \( 1 + (-630. - 1.94e3i)T + (-2.31e7 + 1.68e7i)T^{2} \) |
| 37 | \( 1 + (-8.80e3 - 6.39e3i)T + (2.14e7 + 6.59e7i)T^{2} \) |
| 41 | \( 1 + (-7.66e3 - 5.56e3i)T + (3.58e7 + 1.10e8i)T^{2} \) |
| 43 | \( 1 + 2.29e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + (2.46e3 - 7.58e3i)T + (-1.85e8 - 1.34e8i)T^{2} \) |
| 53 | \( 1 + (-3.57e3 + 1.10e4i)T + (-3.38e8 - 2.45e8i)T^{2} \) |
| 59 | \( 1 + (-3.73e4 - 2.71e4i)T + (2.20e8 + 6.79e8i)T^{2} \) |
| 61 | \( 1 + (1.78e4 - 1.29e4i)T + (2.60e8 - 8.03e8i)T^{2} \) |
| 67 | \( 1 + (8.17e3 + 2.51e4i)T + (-1.09e9 + 7.93e8i)T^{2} \) |
| 71 | \( 1 + (2.14e4 - 6.60e4i)T + (-1.45e9 - 1.06e9i)T^{2} \) |
| 73 | \( 1 + (-1.03e4 + 7.53e3i)T + (6.40e8 - 1.97e9i)T^{2} \) |
| 79 | \( 1 + (-1.81e4 + 5.60e4i)T + (-2.48e9 - 1.80e9i)T^{2} \) |
| 83 | \( 1 + (1.02e4 + 3.14e4i)T + (-3.18e9 + 2.31e9i)T^{2} \) |
| 89 | \( 1 + (5.72e4 - 4.16e4i)T + (1.72e9 - 5.31e9i)T^{2} \) |
| 97 | \( 1 + (2.97e4 - 9.15e4i)T + (-6.94e9 - 5.04e9i)T^{2} \) |
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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
−13.76714858470390970522604306200, −13.06250824109987651765357003706, −11.73464258297100148378181731150, −11.30913071084518893279680266041, −8.801589673903567988339465499754, −7.73482167447974944132223379303, −6.65861175700152596811332362241, −4.56718734482978286157114833776, −2.57857323980574828423270763642, −1.03124087506672633789004946277,
3.35607380817106995313375258825, 4.27520989870538544693332507599, 5.61464798440610015799419372236, 8.092885087320116487453854275504, 8.541784997016346344600623796969, 10.70644339944487968902306808687, 11.06868948086398909252333182115, 12.93816818674737451866151620774, 14.37143211121203236810831577385, 15.18690079802825665337477681334
