| L(s) = 1 | + (1.61 + 2.22i)2-s + (4.81 + 14.8i)3-s + (17.4 − 53.6i)4-s + (−125. − 90.8i)5-s + (−25.1 + 34.6i)6-s + (−357. − 116. i)7-s + (314. − 102. i)8-s + (−196. + 142. i)9-s − 424. i·10-s + (736. − 1.10e3i)11-s + 879.·12-s + (−946. − 1.30e3i)13-s + (−319. − 982. i)14-s + (744. − 2.29e3i)15-s + (−2.18e3 − 1.58e3i)16-s + (−1.03e3 + 1.41e3i)17-s + ⋯ |
| L(s) = 1 | + (0.201 + 0.277i)2-s + (0.178 + 0.549i)3-s + (0.272 − 0.838i)4-s + (−1.00 − 0.726i)5-s + (−0.116 + 0.160i)6-s + (−1.04 − 0.338i)7-s + (0.614 − 0.199i)8-s + (−0.269 + 0.195i)9-s − 0.424i·10-s + (0.553 − 0.832i)11-s + 0.509·12-s + (−0.430 − 0.592i)13-s + (−0.116 − 0.357i)14-s + (0.220 − 0.678i)15-s + (−0.533 − 0.387i)16-s + (−0.209 + 0.288i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.201 + 0.979i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.201 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.717006 - 0.879455i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.717006 - 0.879455i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-4.81 - 14.8i)T \) |
| 11 | \( 1 + (-736. + 1.10e3i)T \) |
| good | 2 | \( 1 + (-1.61 - 2.22i)T + (-19.7 + 60.8i)T^{2} \) |
| 5 | \( 1 + (125. + 90.8i)T + (4.82e3 + 1.48e4i)T^{2} \) |
| 7 | \( 1 + (357. + 116. i)T + (9.51e4 + 6.91e4i)T^{2} \) |
| 13 | \( 1 + (946. + 1.30e3i)T + (-1.49e6 + 4.59e6i)T^{2} \) |
| 17 | \( 1 + (1.03e3 - 1.41e3i)T + (-7.45e6 - 2.29e7i)T^{2} \) |
| 19 | \( 1 + (-1.24e3 + 404. i)T + (3.80e7 - 2.76e7i)T^{2} \) |
| 23 | \( 1 - 9.84e3T + 1.48e8T^{2} \) |
| 29 | \( 1 + (-1.24e4 - 4.04e3i)T + (4.81e8 + 3.49e8i)T^{2} \) |
| 31 | \( 1 + (3.74e4 - 2.71e4i)T + (2.74e8 - 8.44e8i)T^{2} \) |
| 37 | \( 1 + (2.05e4 - 6.31e4i)T + (-2.07e9 - 1.50e9i)T^{2} \) |
| 41 | \( 1 + (-3.84e4 + 1.25e4i)T + (3.84e9 - 2.79e9i)T^{2} \) |
| 43 | \( 1 + 1.19e5iT - 6.32e9T^{2} \) |
| 47 | \( 1 + (6.39e3 + 1.96e4i)T + (-8.72e9 + 6.33e9i)T^{2} \) |
| 53 | \( 1 + (-1.26e5 + 9.16e4i)T + (6.84e9 - 2.10e10i)T^{2} \) |
| 59 | \( 1 + (-1.10e5 + 3.39e5i)T + (-3.41e10 - 2.47e10i)T^{2} \) |
| 61 | \( 1 + (-4.56e4 + 6.28e4i)T + (-1.59e10 - 4.89e10i)T^{2} \) |
| 67 | \( 1 - 3.48e5T + 9.04e10T^{2} \) |
| 71 | \( 1 + (-3.64e5 - 2.64e5i)T + (3.95e10 + 1.21e11i)T^{2} \) |
| 73 | \( 1 + (-5.17e5 - 1.68e5i)T + (1.22e11 + 8.89e10i)T^{2} \) |
| 79 | \( 1 + (5.59e5 + 7.69e5i)T + (-7.51e10 + 2.31e11i)T^{2} \) |
| 83 | \( 1 + (4.14e5 - 5.70e5i)T + (-1.01e11 - 3.10e11i)T^{2} \) |
| 89 | \( 1 + 2.49e5T + 4.96e11T^{2} \) |
| 97 | \( 1 + (-1.04e6 + 7.56e5i)T + (2.57e11 - 7.92e11i)T^{2} \) |
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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
−15.34916384929262338831455894143, −14.08549289681158159419731448516, −12.72315355703174534072682196736, −11.21901646733675904083025752770, −9.984395243945922856568095585670, −8.631959414258379493623183048791, −6.86398669366217129328486331431, −5.20057379961034476468533623953, −3.61239914703035873662336971395, −0.53730699601587328975549325539,
2.59216251773157859517577363564, 3.91926220237248159784380795730, 6.78848221145816072918222762906, 7.53868148494393668351719159538, 9.270863199752776939211375171706, 11.21902004880349768780954350263, 12.14969856289081827911137564539, 12.99926919889552681516468460398, 14.56154995939618337780010478505, 15.74259030852961268921297142168
