| L(s) = 1 | + (−7.21 + 19.8i)3-s + (−40.4 − 229. i)5-s + (−48.6 + 84.2i)7-s + (217. + 182. i)9-s + (17.8 + 30.9i)11-s + (885. + 2.43e3i)13-s + (4.83e3 + 852. i)15-s + (−6.46e3 + 5.42e3i)17-s + (−6.79e3 − 964. i)19-s + (−1.31e3 − 1.57e3i)21-s + (2.97e3 − 1.68e4i)23-s + (−3.62e4 + 1.31e4i)25-s + (−1.85e4 + 1.06e4i)27-s + (−1.38e4 + 1.65e4i)29-s + (4.32e4 + 2.49e4i)31-s + ⋯ |
| L(s) = 1 | + (−0.267 + 0.734i)3-s + (−0.323 − 1.83i)5-s + (−0.141 + 0.245i)7-s + (0.298 + 0.250i)9-s + (0.0134 + 0.0232i)11-s + (0.403 + 1.10i)13-s + (1.43 + 0.252i)15-s + (−1.31 + 1.10i)17-s + (−0.990 − 0.140i)19-s + (−0.142 − 0.169i)21-s + (0.244 − 1.38i)23-s + (−2.31 + 0.844i)25-s + (−0.940 + 0.542i)27-s + (−0.569 + 0.678i)29-s + (1.45 + 0.837i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.667 - 0.744i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.667 - 0.744i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.274238 + 0.614208i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.274238 + 0.614208i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (6.79e3 + 964. i)T \) |
| good | 3 | \( 1 + (7.21 - 19.8i)T + (-558. - 468. i)T^{2} \) |
| 5 | \( 1 + (40.4 + 229. i)T + (-1.46e4 + 5.34e3i)T^{2} \) |
| 7 | \( 1 + (48.6 - 84.2i)T + (-5.88e4 - 1.01e5i)T^{2} \) |
| 11 | \( 1 + (-17.8 - 30.9i)T + (-8.85e5 + 1.53e6i)T^{2} \) |
| 13 | \( 1 + (-885. - 2.43e3i)T + (-3.69e6 + 3.10e6i)T^{2} \) |
| 17 | \( 1 + (6.46e3 - 5.42e3i)T + (4.19e6 - 2.37e7i)T^{2} \) |
| 23 | \( 1 + (-2.97e3 + 1.68e4i)T + (-1.39e8 - 5.06e7i)T^{2} \) |
| 29 | \( 1 + (1.38e4 - 1.65e4i)T + (-1.03e8 - 5.85e8i)T^{2} \) |
| 31 | \( 1 + (-4.32e4 - 2.49e4i)T + (4.43e8 + 7.68e8i)T^{2} \) |
| 37 | \( 1 - 8.69e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 + (545. - 1.50e3i)T + (-3.63e9 - 3.05e9i)T^{2} \) |
| 43 | \( 1 + (-6.98e3 - 3.96e4i)T + (-5.94e9 + 2.16e9i)T^{2} \) |
| 47 | \( 1 + (1.04e5 + 8.76e4i)T + (1.87e9 + 1.06e10i)T^{2} \) |
| 53 | \( 1 + (9.01e4 + 1.58e4i)T + (2.08e10 + 7.58e9i)T^{2} \) |
| 59 | \( 1 + (-1.09e5 - 1.30e5i)T + (-7.32e9 + 4.15e10i)T^{2} \) |
| 61 | \( 1 + (1.25e4 - 7.10e4i)T + (-4.84e10 - 1.76e10i)T^{2} \) |
| 67 | \( 1 + (8.91e4 - 1.06e5i)T + (-1.57e10 - 8.90e10i)T^{2} \) |
| 71 | \( 1 + (-1.07e5 + 1.89e4i)T + (1.20e11 - 4.38e10i)T^{2} \) |
| 73 | \( 1 + (4.32e5 + 1.57e5i)T + (1.15e11 + 9.72e10i)T^{2} \) |
| 79 | \( 1 + (-6.75e4 + 1.85e5i)T + (-1.86e11 - 1.56e11i)T^{2} \) |
| 83 | \( 1 + (-2.09e5 + 3.62e5i)T + (-1.63e11 - 2.83e11i)T^{2} \) |
| 89 | \( 1 + (3.36e5 + 9.25e5i)T + (-3.80e11 + 3.19e11i)T^{2} \) |
| 97 | \( 1 + (7.05e4 + 8.41e4i)T + (-1.44e11 + 8.20e11i)T^{2} \) |
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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
−13.35869892677481552218806242251, −12.68087457166505361116161346144, −11.54151144169049321340235874486, −10.32045575135786873120839301067, −8.951295332136538921788340807576, −8.454947403788859613118608993060, −6.40936374524113027050051051706, −4.72691474409658639508519493753, −4.30251216018889930964303765675, −1.59900678388853529134885480099,
0.26122480787698394905353714079, 2.40305712683438582413723402069, 3.79398605756014819561792332664, 6.09532909286053563722116522682, 6.96057403661070458928926622392, 7.79333265763922004276854906605, 9.734670307304454511467385906983, 10.89952118886035427809889451740, 11.57833481547210139166733659968, 13.02350264974329993372191694139
