| L(s) = 1 | + (2.85 + 3.29i)2-s + (−7.57 − 4.86i)3-s + (1.84 − 12.8i)4-s + (−20.1 + 44.1i)5-s + (−5.58 − 38.8i)6-s + (0.521 + 0.153i)7-s + (165. − 106. i)8-s + (33.6 + 73.6i)9-s + (−203. + 59.6i)10-s + (349. − 403. i)11-s + (−76.5 + 88.3i)12-s + (864. − 253. i)13-s + (0.983 + 2.15i)14-s + (367. − 236. i)15-s + (421. + 123. i)16-s + (−295. − 2.05e3i)17-s + ⋯ |
| L(s) = 1 | + (0.504 + 0.582i)2-s + (−0.485 − 0.312i)3-s + (0.0577 − 0.401i)4-s + (−0.360 + 0.790i)5-s + (−0.0633 − 0.440i)6-s + (0.00402 + 0.00118i)7-s + (0.911 − 0.585i)8-s + (0.138 + 0.303i)9-s + (−0.642 + 0.188i)10-s + (0.871 − 1.00i)11-s + (−0.153 + 0.177i)12-s + (1.41 − 0.416i)13-s + (0.00134 + 0.00293i)14-s + (0.422 − 0.271i)15-s + (0.411 + 0.120i)16-s + (−0.247 − 1.72i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.948 + 0.317i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.948 + 0.317i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.00667 - 0.326986i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.00667 - 0.326986i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (7.57 + 4.86i)T \) |
| 23 | \( 1 + (-1.99e3 + 1.56e3i)T \) |
| good | 2 | \( 1 + (-2.85 - 3.29i)T + (-4.55 + 31.6i)T^{2} \) |
| 5 | \( 1 + (20.1 - 44.1i)T + (-2.04e3 - 2.36e3i)T^{2} \) |
| 7 | \( 1 + (-0.521 - 0.153i)T + (1.41e4 + 9.08e3i)T^{2} \) |
| 11 | \( 1 + (-349. + 403. i)T + (-2.29e4 - 1.59e5i)T^{2} \) |
| 13 | \( 1 + (-864. + 253. i)T + (3.12e5 - 2.00e5i)T^{2} \) |
| 17 | \( 1 + (295. + 2.05e3i)T + (-1.36e6 + 4.00e5i)T^{2} \) |
| 19 | \( 1 + (135. - 940. i)T + (-2.37e6 - 6.97e5i)T^{2} \) |
| 29 | \( 1 + (-1.02e3 - 7.12e3i)T + (-1.96e7 + 5.77e6i)T^{2} \) |
| 31 | \( 1 + (5.62e3 - 3.61e3i)T + (1.18e7 - 2.60e7i)T^{2} \) |
| 37 | \( 1 + (3.86e3 + 8.46e3i)T + (-4.54e7 + 5.24e7i)T^{2} \) |
| 41 | \( 1 + (-7.08e3 + 1.55e4i)T + (-7.58e7 - 8.75e7i)T^{2} \) |
| 43 | \( 1 + (-1.69e4 - 1.08e4i)T + (6.10e7 + 1.33e8i)T^{2} \) |
| 47 | \( 1 + 1.83e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + (3.83e3 + 1.12e3i)T + (3.51e8 + 2.26e8i)T^{2} \) |
| 59 | \( 1 + (1.51e4 - 4.45e3i)T + (6.01e8 - 3.86e8i)T^{2} \) |
| 61 | \( 1 + (-9.86e3 + 6.34e3i)T + (3.50e8 - 7.68e8i)T^{2} \) |
| 67 | \( 1 + (-3.01e4 - 3.47e4i)T + (-1.92e8 + 1.33e9i)T^{2} \) |
| 71 | \( 1 + (1.36e4 + 1.57e4i)T + (-2.56e8 + 1.78e9i)T^{2} \) |
| 73 | \( 1 + (5.44e3 - 3.79e4i)T + (-1.98e9 - 5.84e8i)T^{2} \) |
| 79 | \( 1 + (-3.78e4 + 1.11e4i)T + (2.58e9 - 1.66e9i)T^{2} \) |
| 83 | \( 1 + (-1.08e4 - 2.36e4i)T + (-2.57e9 + 2.97e9i)T^{2} \) |
| 89 | \( 1 + (-9.55e4 - 6.14e4i)T + (2.31e9 + 5.07e9i)T^{2} \) |
| 97 | \( 1 + (-2.61e4 + 5.71e4i)T + (-5.62e9 - 6.48e9i)T^{2} \) |
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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
−14.02110594527410934246499821608, −12.79468201390182505397894368805, −11.16890781175417423997741601991, −10.81496250498572645061708295530, −8.970656008400698899723138861501, −7.23641554333885009723762610697, −6.43703379468876099051031635082, −5.30266458628988883712035596853, −3.51383844967333851983715993491, −0.983460271913952543017030967289,
1.53281135807337127941976704328, 3.83422994879613311020225549333, 4.58277337733497246073548487702, 6.38139135856988544025898225218, 8.067998854676250711019418180574, 9.247126895462922742586805723488, 10.87193587244781525055501666239, 11.66042015005541418220821191056, 12.63895391388678154746745357307, 13.35433811196278817712211051458
