| L(s) = 1 | + (7.35 + 12.7i)2-s + (42.7 + 74.0i)3-s + (147. − 255. i)4-s + (−1.10e3 − 1.91e3i)5-s + (−629. + 1.08e3i)6-s − 1.21e4·7-s + 1.18e4·8-s + (6.18e3 − 1.07e4i)9-s + (1.62e4 − 2.82e4i)10-s − 3.31e4·11-s + 2.52e4·12-s + (8.64e3 − 1.49e4i)13-s + (−8.96e4 − 1.55e5i)14-s + (9.46e4 − 1.63e5i)15-s + (1.18e4 + 2.05e4i)16-s + (−5.34e4 − 9.25e4i)17-s + ⋯ |
| L(s) = 1 | + (0.325 + 0.563i)2-s + (0.304 + 0.527i)3-s + (0.288 − 0.499i)4-s + (−0.792 − 1.37i)5-s + (−0.198 + 0.343i)6-s − 1.91·7-s + 1.02·8-s + (0.314 − 0.544i)9-s + (0.515 − 0.892i)10-s − 0.682·11-s + 0.351·12-s + (0.0839 − 0.145i)13-s + (−0.623 − 1.08i)14-s + (0.482 − 0.836i)15-s + (0.0451 + 0.0782i)16-s + (−0.155 − 0.268i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00511 + 0.999i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.00511 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.900063 - 0.904674i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.900063 - 0.904674i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 + (-5.39e5 - 1.76e5i)T \) |
| good | 2 | \( 1 + (-7.35 - 12.7i)T + (-256 + 443. i)T^{2} \) |
| 3 | \( 1 + (-42.7 - 74.0i)T + (-9.84e3 + 1.70e4i)T^{2} \) |
| 5 | \( 1 + (1.10e3 + 1.91e3i)T + (-9.76e5 + 1.69e6i)T^{2} \) |
| 7 | \( 1 + 1.21e4T + 4.03e7T^{2} \) |
| 11 | \( 1 + 3.31e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + (-8.64e3 + 1.49e4i)T + (-5.30e9 - 9.18e9i)T^{2} \) |
| 17 | \( 1 + (5.34e4 + 9.25e4i)T + (-5.92e10 + 1.02e11i)T^{2} \) |
| 23 | \( 1 + (-3.94e5 + 6.83e5i)T + (-9.00e11 - 1.55e12i)T^{2} \) |
| 29 | \( 1 + (1.69e6 - 2.94e6i)T + (-7.25e12 - 1.25e13i)T^{2} \) |
| 31 | \( 1 + 4.14e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 2.25e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + (1.30e7 + 2.26e7i)T + (-1.63e14 + 2.83e14i)T^{2} \) |
| 43 | \( 1 + (1.02e7 + 1.76e7i)T + (-2.51e14 + 4.35e14i)T^{2} \) |
| 47 | \( 1 + (-1.28e6 + 2.22e6i)T + (-5.59e14 - 9.69e14i)T^{2} \) |
| 53 | \( 1 + (-3.08e7 + 5.33e7i)T + (-1.64e15 - 2.85e15i)T^{2} \) |
| 59 | \( 1 + (-6.97e7 - 1.20e8i)T + (-4.33e15 + 7.50e15i)T^{2} \) |
| 61 | \( 1 + (-3.71e7 + 6.43e7i)T + (-5.84e15 - 1.01e16i)T^{2} \) |
| 67 | \( 1 + (-2.53e7 + 4.39e7i)T + (-1.36e16 - 2.35e16i)T^{2} \) |
| 71 | \( 1 + (-1.20e8 - 2.08e8i)T + (-2.29e16 + 3.97e16i)T^{2} \) |
| 73 | \( 1 + (1.19e8 + 2.06e8i)T + (-2.94e16 + 5.09e16i)T^{2} \) |
| 79 | \( 1 + (1.96e8 + 3.39e8i)T + (-5.99e16 + 1.03e17i)T^{2} \) |
| 83 | \( 1 - 7.00e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + (1.91e8 - 3.31e8i)T + (-1.75e17 - 3.03e17i)T^{2} \) |
| 97 | \( 1 + (1.06e8 + 1.84e8i)T + (-3.80e17 + 6.58e17i)T^{2} \) |
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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
−16.08203502744954971766191268433, −15.20955956090513511039840331423, −13.34041983242003893430562138581, −12.33239544342047189559268330349, −10.16829286233725871975359696992, −9.011285995715913579323191196961, −7.05344652422939058242247081960, −5.32415981007873535844657960862, −3.66441402582783497598749260734, −0.50993629229935104583066747133,
2.64663088300176729082883726178, 3.52260531701082271069249717976, 6.78953318454530214665669578159, 7.64075570709122644732901558811, 10.12096962908271811480510961078, 11.36115123379669483908249377082, 12.79639807554055665738917208161, 13.58397708920290222777504745159, 15.49691600798628187744306715489, 16.37371946431253279158995765522
