| L(s) = 1 | + (−9.90 + 6.36i)2-s + (2.21 − 15.4i)3-s + (30.9 − 67.8i)4-s + (54.8 + 186. i)5-s + (76.2 + 166. i)6-s + (134. − 116. i)7-s + (17.6 + 122. i)8-s + (−233. − 68.4i)9-s + (−1.73e3 − 1.50e3i)10-s + (−263. + 410. i)11-s + (−977. − 628. i)12-s + (2.62e3 − 3.03e3i)13-s + (−591. + 2.01e3i)14-s + (3.00e3 − 432. i)15-s + (2.16e3 + 2.50e3i)16-s + (1.68e3 − 769. i)17-s + ⋯ |
| L(s) = 1 | + (−1.23 + 0.795i)2-s + (0.0821 − 0.571i)3-s + (0.483 − 1.05i)4-s + (0.439 + 1.49i)5-s + (0.352 + 0.772i)6-s + (0.392 − 0.340i)7-s + (0.0345 + 0.240i)8-s + (−0.319 − 0.0939i)9-s + (−1.73 − 1.50i)10-s + (−0.198 + 0.308i)11-s + (−0.565 − 0.363i)12-s + (1.19 − 1.38i)13-s + (−0.215 + 0.734i)14-s + (0.890 − 0.128i)15-s + (0.529 + 0.610i)16-s + (0.343 − 0.156i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.216 - 0.976i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.216 - 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.850983 + 0.683159i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.850983 + 0.683159i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-2.21 + 15.4i)T \) |
| 23 | \( 1 + (-1.10e4 - 5.03e3i)T \) |
| good | 2 | \( 1 + (9.90 - 6.36i)T + (26.5 - 58.2i)T^{2} \) |
| 5 | \( 1 + (-54.8 - 186. i)T + (-1.31e4 + 8.44e3i)T^{2} \) |
| 7 | \( 1 + (-134. + 116. i)T + (1.67e4 - 1.16e5i)T^{2} \) |
| 11 | \( 1 + (263. - 410. i)T + (-7.35e5 - 1.61e6i)T^{2} \) |
| 13 | \( 1 + (-2.62e3 + 3.03e3i)T + (-6.86e5 - 4.77e6i)T^{2} \) |
| 17 | \( 1 + (-1.68e3 + 769. i)T + (1.58e7 - 1.82e7i)T^{2} \) |
| 19 | \( 1 + (4.99e3 + 2.28e3i)T + (3.08e7 + 3.55e7i)T^{2} \) |
| 29 | \( 1 + (-1.01e4 - 2.22e4i)T + (-3.89e8 + 4.49e8i)T^{2} \) |
| 31 | \( 1 + (-5.43e3 - 3.78e4i)T + (-8.51e8 + 2.50e8i)T^{2} \) |
| 37 | \( 1 + (7.86e3 - 2.67e4i)T + (-2.15e9 - 1.38e9i)T^{2} \) |
| 41 | \( 1 + (-7.34e4 + 2.15e4i)T + (3.99e9 - 2.56e9i)T^{2} \) |
| 43 | \( 1 + (-4.66e4 - 6.70e3i)T + (6.06e9 + 1.78e9i)T^{2} \) |
| 47 | \( 1 - 1.55e5T + 1.07e10T^{2} \) |
| 53 | \( 1 + (1.06e5 - 9.26e4i)T + (3.15e9 - 2.19e10i)T^{2} \) |
| 59 | \( 1 + (1.17e5 - 1.35e5i)T + (-6.00e9 - 4.17e10i)T^{2} \) |
| 61 | \( 1 + (-2.59e5 + 3.72e4i)T + (4.94e10 - 1.45e10i)T^{2} \) |
| 67 | \( 1 + (2.88e4 + 4.48e4i)T + (-3.75e10 + 8.22e10i)T^{2} \) |
| 71 | \( 1 + (1.67e5 - 1.07e5i)T + (5.32e10 - 1.16e11i)T^{2} \) |
| 73 | \( 1 + (-5.45e4 + 1.19e5i)T + (-9.91e10 - 1.14e11i)T^{2} \) |
| 79 | \( 1 + (7.20e5 + 6.24e5i)T + (3.45e10 + 2.40e11i)T^{2} \) |
| 83 | \( 1 + (2.70e4 - 9.19e4i)T + (-2.75e11 - 1.76e11i)T^{2} \) |
| 89 | \( 1 + (-6.80e5 - 9.78e4i)T + (4.76e11 + 1.40e11i)T^{2} \) |
| 97 | \( 1 + (-4.36e5 - 1.48e6i)T + (-7.00e11 + 4.50e11i)T^{2} \) |
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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
−14.03355744773538681161008148948, −12.81186645332109820584514909041, −10.82315877114880470429730428565, −10.46060731541517245681379487390, −8.893900022050095562466332913789, −7.72124434477201217448967604140, −6.93912313218196359774094233847, −5.90702031910258694594352166578, −3.05051749849228315999106634500, −1.11895248436420689569774679597,
0.821085880191560172872126912459, 2.07307235736410012935837094950, 4.28634681587975303327037717273, 5.76821104928138286120794562152, 8.247829577755552547785281429678, 8.863241576379923750934002701045, 9.596423333030702979313287175321, 10.89969381234284402500579152236, 11.78707622964354217390160563022, 13.00979652735160081161908186621
