| L(s) = 1 | + (−16.6 + 45.8i)3-s + (9.11 + 51.7i)5-s + (−320. + 554. i)7-s + (−1.26e3 − 1.06e3i)9-s + (1.05e3 + 1.82e3i)11-s + (160. + 440. i)13-s + (−2.52e3 − 445. i)15-s + (2.15e3 − 1.80e3i)17-s + (583. + 6.83e3i)19-s + (−2.01e4 − 2.39e4i)21-s + (2.98e3 − 1.69e4i)23-s + (1.20e4 − 4.40e3i)25-s + (3.91e4 − 2.25e4i)27-s + (−238. + 284. i)29-s + (−2.39e4 − 1.38e4i)31-s + ⋯ |
| L(s) = 1 | + (−0.618 + 1.69i)3-s + (0.0729 + 0.413i)5-s + (−0.934 + 1.61i)7-s + (−1.73 − 1.45i)9-s + (0.789 + 1.36i)11-s + (0.0730 + 0.200i)13-s + (−0.747 − 0.131i)15-s + (0.437 − 0.367i)17-s + (0.0850 + 0.996i)19-s + (−2.17 − 2.58i)21-s + (0.245 − 1.39i)23-s + (0.773 − 0.281i)25-s + (1.98 − 1.14i)27-s + (−0.00979 + 0.0116i)29-s + (−0.804 − 0.464i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.575 + 0.817i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.575 + 0.817i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.460574 - 0.887724i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.460574 - 0.887724i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (-583. - 6.83e3i)T \) |
| good | 3 | \( 1 + (16.6 - 45.8i)T + (-558. - 468. i)T^{2} \) |
| 5 | \( 1 + (-9.11 - 51.7i)T + (-1.46e4 + 5.34e3i)T^{2} \) |
| 7 | \( 1 + (320. - 554. i)T + (-5.88e4 - 1.01e5i)T^{2} \) |
| 11 | \( 1 + (-1.05e3 - 1.82e3i)T + (-8.85e5 + 1.53e6i)T^{2} \) |
| 13 | \( 1 + (-160. - 440. i)T + (-3.69e6 + 3.10e6i)T^{2} \) |
| 17 | \( 1 + (-2.15e3 + 1.80e3i)T + (4.19e6 - 2.37e7i)T^{2} \) |
| 23 | \( 1 + (-2.98e3 + 1.69e4i)T + (-1.39e8 - 5.06e7i)T^{2} \) |
| 29 | \( 1 + (238. - 284. i)T + (-1.03e8 - 5.85e8i)T^{2} \) |
| 31 | \( 1 + (2.39e4 + 1.38e4i)T + (4.43e8 + 7.68e8i)T^{2} \) |
| 37 | \( 1 - 3.67e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 + (-1.68e4 + 4.62e4i)T + (-3.63e9 - 3.05e9i)T^{2} \) |
| 43 | \( 1 + (-1.84e4 - 1.04e5i)T + (-5.94e9 + 2.16e9i)T^{2} \) |
| 47 | \( 1 + (-3.52e4 - 2.96e4i)T + (1.87e9 + 1.06e10i)T^{2} \) |
| 53 | \( 1 + (-2.22e5 - 3.92e4i)T + (2.08e10 + 7.58e9i)T^{2} \) |
| 59 | \( 1 + (-3.68e4 - 4.39e4i)T + (-7.32e9 + 4.15e10i)T^{2} \) |
| 61 | \( 1 + (4.69e4 - 2.66e5i)T + (-4.84e10 - 1.76e10i)T^{2} \) |
| 67 | \( 1 + (-9.73e4 + 1.16e5i)T + (-1.57e10 - 8.90e10i)T^{2} \) |
| 71 | \( 1 + (4.90e5 - 8.65e4i)T + (1.20e11 - 4.38e10i)T^{2} \) |
| 73 | \( 1 + (1.37e5 + 4.99e4i)T + (1.15e11 + 9.72e10i)T^{2} \) |
| 79 | \( 1 + (1.05e4 - 2.88e4i)T + (-1.86e11 - 1.56e11i)T^{2} \) |
| 83 | \( 1 + (-4.43e4 + 7.68e4i)T + (-1.63e11 - 2.83e11i)T^{2} \) |
| 89 | \( 1 + (-4.70e5 - 1.29e6i)T + (-3.80e11 + 3.19e11i)T^{2} \) |
| 97 | \( 1 + (-1.49e5 - 1.78e5i)T + (-1.44e11 + 8.20e11i)T^{2} \) |
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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
−14.80144481221682956005987062497, −12.43789736922177072203680463421, −11.83954643073072019468002555789, −10.44560676900486863425303379030, −9.616710857981081040617863524429, −8.917081931600828983926320999184, −6.53715126881976206572147187154, −5.52156295073788814683385529939, −4.21456056795372937638378962227, −2.77393087806476351919191468191,
0.47617671926104188409250780252, 1.18594002911520235855734652497, 3.44884631964028597249143995914, 5.64163409078499027991836553883, 6.75636713005567818693115531494, 7.48822939504750046256250069967, 8.938654387034411153187771772697, 10.68769216184351093640896056848, 11.58625547139490819706328954067, 12.87146161394992561259109430673
