L(s) = 1 | + (2.24 + 5.19i)2-s + (−15.5 + 1.52i)3-s + (−21.9 + 23.2i)4-s − 25i·5-s + (−42.7 − 77.1i)6-s − 85.2i·7-s + (−170. − 61.6i)8-s + (238. − 47.3i)9-s + (129. − 56.0i)10-s − 246.·11-s + (304. − 394. i)12-s + 940.·13-s + (442. − 191. i)14-s + (38.1 + 387. i)15-s + (−61.4 − 1.02e3i)16-s − 639. i·17-s + ⋯ |
L(s) = 1 | + (0.396 + 0.918i)2-s + (−0.995 + 0.0979i)3-s + (−0.685 + 0.728i)4-s − 0.447i·5-s + (−0.484 − 0.874i)6-s − 0.657i·7-s + (−0.940 − 0.340i)8-s + (0.980 − 0.194i)9-s + (0.410 − 0.177i)10-s − 0.613·11-s + (0.610 − 0.791i)12-s + 1.54·13-s + (0.603 − 0.260i)14-s + (0.0437 + 0.445i)15-s + (−0.0600 − 0.998i)16-s − 0.536i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.791 + 0.610i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.791 + 0.610i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.896948 - 0.305876i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.896948 - 0.305876i\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-2.24 - 5.19i)T \) |
| 3 | \( 1 + (15.5 - 1.52i)T \) |
| 5 | \( 1 + 25iT \) |
good | 7 | \( 1 + 85.2iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 246.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 940.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 639. iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 2.05e3iT - 2.47e6T^{2} \) |
| 23 | \( 1 + 1.54e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 2.38e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 + 5.33e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 + 1.37e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.92e4iT - 1.15e8T^{2} \) |
| 43 | \( 1 - 1.26e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 5.17e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.80e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 9.34e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.68e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.96e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 4.61e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 5.44e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 2.81e4iT - 3.07e9T^{2} \) |
| 83 | \( 1 - 1.17e5T + 3.93e9T^{2} \) |
| 89 | \( 1 + 6.72e4iT - 5.58e9T^{2} \) |
| 97 | \( 1 - 1.53e3T + 8.58e9T^{2} \) |
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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
−13.72631175982573708121827542675, −13.14378307211439262187649596233, −11.84958364927404038527776526524, −10.63028599206294792853026983330, −9.094236853088366894375634826809, −7.62992356836382932210427058258, −6.39900483087755163547276362770, −5.21614374415484436938329557737, −3.99320433457564486162313237894, −0.47286410771061709143370441706,
1.61284424841117734462475162106, 3.64247753377906044586996849089, 5.37457992856743604765966457027, 6.32035782780157166530071399862, 8.442996807799369353914564841835, 10.14536854191531435249401834535, 10.86212935994925899339534967500, 11.92624099253823044456309148441, 12.78680885454613908973711777530, 13.91156050182078693104407285647