| L(s) = 1 | + (7.59 + 8.76i)2-s + (13.1 + 8.42i)3-s + (−10.0 + 69.7i)4-s + (−181. − 83.0i)5-s + (25.7 + 178. i)6-s + (−148. + 505. i)7-s + (−63.2 + 40.6i)8-s + (100. + 221. i)9-s + (−653. − 2.22e3i)10-s + (−1.04e3 − 905. i)11-s + (−719. + 830. i)12-s + (−1.61e3 + 475. i)13-s + (−5.56e3 + 2.53e3i)14-s + (−1.68e3 − 2.62e3i)15-s + (3.49e3 + 1.02e3i)16-s + (−7.79e3 + 1.12e3i)17-s + ⋯ |
| L(s) = 1 | + (0.949 + 1.09i)2-s + (0.485 + 0.312i)3-s + (−0.156 + 1.09i)4-s + (−1.45 − 0.664i)5-s + (0.119 + 0.828i)6-s + (−0.432 + 1.47i)7-s + (−0.123 + 0.0794i)8-s + (0.138 + 0.303i)9-s + (−0.653 − 2.22i)10-s + (−0.784 − 0.680i)11-s + (−0.416 + 0.480i)12-s + (−0.737 + 0.216i)13-s + (−2.02 + 0.925i)14-s + (−0.499 − 0.776i)15-s + (0.852 + 0.250i)16-s + (−1.58 + 0.227i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.808 + 0.588i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.808 + 0.588i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.395761 - 1.21730i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.395761 - 1.21730i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-13.1 - 8.42i)T \) |
| 23 | \( 1 + (-8.93e3 + 8.26e3i)T \) |
| good | 2 | \( 1 + (-7.59 - 8.76i)T + (-9.10 + 63.3i)T^{2} \) |
| 5 | \( 1 + (181. + 83.0i)T + (1.02e4 + 1.18e4i)T^{2} \) |
| 7 | \( 1 + (148. - 505. i)T + (-9.89e4 - 6.36e4i)T^{2} \) |
| 11 | \( 1 + (1.04e3 + 905. i)T + (2.52e5 + 1.75e6i)T^{2} \) |
| 13 | \( 1 + (1.61e3 - 475. i)T + (4.06e6 - 2.60e6i)T^{2} \) |
| 17 | \( 1 + (7.79e3 - 1.12e3i)T + (2.31e7 - 6.80e6i)T^{2} \) |
| 19 | \( 1 + (-3.65e3 - 525. i)T + (4.51e7 + 1.32e7i)T^{2} \) |
| 29 | \( 1 + (-5.09e3 - 3.54e4i)T + (-5.70e8 + 1.67e8i)T^{2} \) |
| 31 | \( 1 + (4.22e4 - 2.71e4i)T + (3.68e8 - 8.07e8i)T^{2} \) |
| 37 | \( 1 + (-7.56e4 + 3.45e4i)T + (1.68e9 - 1.93e9i)T^{2} \) |
| 41 | \( 1 + (-7.64e3 + 1.67e4i)T + (-3.11e9 - 3.58e9i)T^{2} \) |
| 43 | \( 1 + (8.13e4 - 1.26e5i)T + (-2.62e9 - 5.75e9i)T^{2} \) |
| 47 | \( 1 - 6.11e4T + 1.07e10T^{2} \) |
| 53 | \( 1 + (2.42e3 - 8.26e3i)T + (-1.86e10 - 1.19e10i)T^{2} \) |
| 59 | \( 1 + (-9.62e4 + 2.82e4i)T + (3.54e10 - 2.28e10i)T^{2} \) |
| 61 | \( 1 + (-2.41e4 - 3.75e4i)T + (-2.14e10 + 4.68e10i)T^{2} \) |
| 67 | \( 1 + (7.06e4 - 6.12e4i)T + (1.28e10 - 8.95e10i)T^{2} \) |
| 71 | \( 1 + (2.10e4 + 2.42e4i)T + (-1.82e10 + 1.26e11i)T^{2} \) |
| 73 | \( 1 + (3.75e4 - 2.61e5i)T + (-1.45e11 - 4.26e10i)T^{2} \) |
| 79 | \( 1 + (9.53e4 + 3.24e5i)T + (-2.04e11 + 1.31e11i)T^{2} \) |
| 83 | \( 1 + (6.56e5 - 2.99e5i)T + (2.14e11 - 2.47e11i)T^{2} \) |
| 89 | \( 1 + (2.11e5 - 3.28e5i)T + (-2.06e11 - 4.52e11i)T^{2} \) |
| 97 | \( 1 + (-7.74e5 - 3.53e5i)T + (5.45e11 + 6.29e11i)T^{2} \) |
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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
−14.62015237950334640262738266078, −13.02124072740452249196986175019, −12.49858149685699680999558962099, −11.11370890020318764472788044605, −9.027207698085210182868202525695, −8.264980752248797392813934430331, −7.02633036270967839771825347239, −5.42011268633259976078516838169, −4.48024923466977065694280878704, −3.02597777230744077446292123590,
0.33224272161495398460977065122, 2.51989514638824458942908597152, 3.69194244996413594526008348261, 4.54014390045595229313076430785, 7.15576213848145589118205044633, 7.69970185209531097201931662538, 9.911987320757989767941909554961, 10.95891825716713047296626645353, 11.72354541610650163839378944461, 13.02226671007644894959494046846
