| L(s) = 1 | + (4.32 − 8.97i)2-s + (26.7 + 21.3i)3-s + (17.8 + 22.4i)4-s + (−387. − 186. i)5-s + (307. − 148. i)6-s + (921. − 1.15e3i)7-s + (1.52e3 − 347. i)8-s + (−225. − 987. i)9-s + (−3.35e3 + 2.67e3i)10-s + (5.68e3 + 1.29e3i)11-s + 982. i·12-s + (1.41e3 − 6.22e3i)13-s + (−6.39e3 − 1.32e4i)14-s + (−6.39e3 − 1.32e4i)15-s + (2.64e3 − 1.15e4i)16-s + 8.33e3i·17-s + ⋯ |
| L(s) = 1 | + (0.382 − 0.793i)2-s + (0.572 + 0.456i)3-s + (0.139 + 0.175i)4-s + (−1.38 − 0.667i)5-s + (0.581 − 0.280i)6-s + (1.01 − 1.27i)7-s + (1.05 − 0.239i)8-s + (−0.103 − 0.451i)9-s + (−1.06 + 0.845i)10-s + (1.28 + 0.293i)11-s + 0.164i·12-s + (0.179 − 0.785i)13-s + (−0.622 − 1.29i)14-s + (−0.489 − 1.01i)15-s + (0.161 − 0.707i)16-s + 0.411i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.149 + 0.988i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.149 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.86597 - 1.60575i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.86597 - 1.60575i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 29 | \( 1 + (8.73e3 - 1.31e5i)T \) |
| good | 2 | \( 1 + (-4.32 + 8.97i)T + (-79.8 - 100. i)T^{2} \) |
| 3 | \( 1 + (-26.7 - 21.3i)T + (486. + 2.13e3i)T^{2} \) |
| 5 | \( 1 + (387. + 186. i)T + (4.87e4 + 6.10e4i)T^{2} \) |
| 7 | \( 1 + (-921. + 1.15e3i)T + (-1.83e5 - 8.02e5i)T^{2} \) |
| 11 | \( 1 + (-5.68e3 - 1.29e3i)T + (1.75e7 + 8.45e6i)T^{2} \) |
| 13 | \( 1 + (-1.41e3 + 6.22e3i)T + (-5.65e7 - 2.72e7i)T^{2} \) |
| 17 | \( 1 - 8.33e3iT - 4.10e8T^{2} \) |
| 19 | \( 1 + (2.14e4 - 1.71e4i)T + (1.98e8 - 8.71e8i)T^{2} \) |
| 23 | \( 1 + (7.89e4 - 3.80e4i)T + (2.12e9 - 2.66e9i)T^{2} \) |
| 31 | \( 1 + (4.87e3 - 1.01e4i)T + (-1.71e10 - 2.15e10i)T^{2} \) |
| 37 | \( 1 + (-1.84e5 + 4.20e4i)T + (8.55e10 - 4.11e10i)T^{2} \) |
| 41 | \( 1 + 1.50e5iT - 1.94e11T^{2} \) |
| 43 | \( 1 + (1.48e5 + 3.08e5i)T + (-1.69e11 + 2.12e11i)T^{2} \) |
| 47 | \( 1 + (-7.93e5 - 1.81e5i)T + (4.56e11 + 2.19e11i)T^{2} \) |
| 53 | \( 1 + (-1.44e6 - 6.95e5i)T + (7.32e11 + 9.18e11i)T^{2} \) |
| 59 | \( 1 + 2.35e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + (-1.11e6 - 8.91e5i)T + (6.99e11 + 3.06e12i)T^{2} \) |
| 67 | \( 1 + (-5.92e5 - 2.59e6i)T + (-5.46e12 + 2.62e12i)T^{2} \) |
| 71 | \( 1 + (1.30e6 - 5.70e6i)T + (-8.19e12 - 3.94e12i)T^{2} \) |
| 73 | \( 1 + (1.54e6 + 3.20e6i)T + (-6.88e12 + 8.63e12i)T^{2} \) |
| 79 | \( 1 + (-1.47e6 + 3.36e5i)T + (1.73e13 - 8.33e12i)T^{2} \) |
| 83 | \( 1 + (-6.72e5 - 8.43e5i)T + (-6.03e12 + 2.64e13i)T^{2} \) |
| 89 | \( 1 + (3.93e6 - 8.16e6i)T + (-2.75e13 - 3.45e13i)T^{2} \) |
| 97 | \( 1 + (5.17e5 - 4.12e5i)T + (1.79e13 - 7.87e13i)T^{2} \) |
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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
−15.11914289385612461152097579732, −14.03819357625342267828920896523, −12.43939315682311624675283310479, −11.62105532390046856041418437105, −10.42974784406438351897313935844, −8.486116914205224225730219544676, −7.46577462560240897054301121904, −4.08643515078854705510079326495, −3.89438088851866785684287008936, −1.17431975486686740201146773588,
2.12345856060212914483548582498, 4.41991821799433377842775362394, 6.34735454509985695021260765509, 7.67288683651029616559109045333, 8.610198561259470236875280174206, 11.17682503892567230029162643143, 11.86365423563556367359891534239, 13.97821966993396581402033532522, 14.69239896116886886391536370776, 15.44582869727316600439711511112
