| L(s) = 1 | + (−2.76 + 1.77i)2-s + (−1.28 + 8.90i)3-s + (−8.82 + 19.3i)4-s + (−80.1 + 23.5i)5-s + (−12.2 − 26.8i)6-s + (−77.8 − 89.8i)7-s + (−24.8 − 172. i)8-s + (−77.7 − 22.8i)9-s + (179. − 207. i)10-s + (635. + 408. i)11-s + (−160. − 103. i)12-s + (−201. + 232. i)13-s + (374. + 109. i)14-s + (−107. − 744. i)15-s + (−69.6 − 80.4i)16-s + (74.3 + 162. i)17-s + ⋯ |
| L(s) = 1 | + (−0.487 + 0.313i)2-s + (−0.0821 + 0.571i)3-s + (−0.275 + 0.603i)4-s + (−1.43 + 0.421i)5-s + (−0.139 − 0.304i)6-s + (−0.600 − 0.693i)7-s + (−0.137 − 0.955i)8-s + (−0.319 − 0.0939i)9-s + (0.567 − 0.655i)10-s + (1.58 + 1.01i)11-s + (−0.322 − 0.207i)12-s + (−0.330 + 0.381i)13-s + (0.510 + 0.149i)14-s + (−0.122 − 0.854i)15-s + (−0.0680 − 0.0785i)16-s + (0.0623 + 0.136i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.548 + 0.835i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.548 + 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.196417 - 0.106016i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.196417 - 0.106016i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.28 - 8.90i)T \) |
| 23 | \( 1 + (-836. - 2.39e3i)T \) |
| good | 2 | \( 1 + (2.76 - 1.77i)T + (13.2 - 29.1i)T^{2} \) |
| 5 | \( 1 + (80.1 - 23.5i)T + (2.62e3 - 1.68e3i)T^{2} \) |
| 7 | \( 1 + (77.8 + 89.8i)T + (-2.39e3 + 1.66e4i)T^{2} \) |
| 11 | \( 1 + (-635. - 408. i)T + (6.69e4 + 1.46e5i)T^{2} \) |
| 13 | \( 1 + (201. - 232. i)T + (-5.28e4 - 3.67e5i)T^{2} \) |
| 17 | \( 1 + (-74.3 - 162. i)T + (-9.29e5 + 1.07e6i)T^{2} \) |
| 19 | \( 1 + (-533. + 1.16e3i)T + (-1.62e6 - 1.87e6i)T^{2} \) |
| 29 | \( 1 + (3.34e3 + 7.32e3i)T + (-1.34e7 + 1.55e7i)T^{2} \) |
| 31 | \( 1 + (868. + 6.03e3i)T + (-2.74e7 + 8.06e6i)T^{2} \) |
| 37 | \( 1 + (1.45e4 + 4.28e3i)T + (5.83e7 + 3.74e7i)T^{2} \) |
| 41 | \( 1 + (1.61e4 - 4.74e3i)T + (9.74e7 - 6.26e7i)T^{2} \) |
| 43 | \( 1 + (-467. + 3.25e3i)T + (-1.41e8 - 4.14e7i)T^{2} \) |
| 47 | \( 1 - 2.30e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + (-1.36e4 - 1.57e4i)T + (-5.95e7 + 4.13e8i)T^{2} \) |
| 59 | \( 1 + (5.27e3 - 6.08e3i)T + (-1.01e8 - 7.07e8i)T^{2} \) |
| 61 | \( 1 + (3.07e3 + 2.13e4i)T + (-8.10e8 + 2.37e8i)T^{2} \) |
| 67 | \( 1 + (3.36e3 - 2.16e3i)T + (5.60e8 - 1.22e9i)T^{2} \) |
| 71 | \( 1 + (-3.44e4 + 2.21e4i)T + (7.49e8 - 1.64e9i)T^{2} \) |
| 73 | \( 1 + (2.07e4 - 4.54e4i)T + (-1.35e9 - 1.56e9i)T^{2} \) |
| 79 | \( 1 + (-2.22e4 + 2.57e4i)T + (-4.37e8 - 3.04e9i)T^{2} \) |
| 83 | \( 1 + (6.49e4 + 1.90e4i)T + (3.31e9 + 2.12e9i)T^{2} \) |
| 89 | \( 1 + (-9.64e3 + 6.70e4i)T + (-5.35e9 - 1.57e9i)T^{2} \) |
| 97 | \( 1 + (-4.78e4 + 1.40e4i)T + (7.22e9 - 4.64e9i)T^{2} \) |
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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
−13.64617006909803309891577719082, −12.15591963480303904319298268590, −11.49152994230055701248880732160, −9.890066419040939236429346546535, −9.018547700052860478477848081717, −7.49217016280274754719792338638, −6.90002046535915680190831253466, −4.20141206579055620723096343072, −3.65059949459857216725416630335, −0.14503683398685786766799275244,
1.16027909349270932693617201282, 3.46666159341542575019240939317, 5.35539770265095021921175301863, 6.81866465740412777956087674144, 8.487169658809606136988476794120, 8.997356400950714764888507946925, 10.67636414480048247983849921341, 11.85952158030858931866561210971, 12.36039606022041612371917107476, 14.01434599060168163382574236413
