| L(s) = 1 | + (−9.04 + 5.81i)2-s + (−1.28 + 8.90i)3-s + (34.7 − 76.0i)4-s + (47.9 − 14.0i)5-s + (−40.2 − 88.0i)6-s + (−47.4 − 54.7i)7-s + (79.0 + 549. i)8-s + (−77.7 − 22.8i)9-s + (−352. + 406. i)10-s + (206. + 132. i)11-s + (633. + 406. i)12-s + (−751. + 866. i)13-s + (747. + 219. i)14-s + (64.0 + 445. i)15-s + (−2.15e3 − 2.48e3i)16-s + (−130. − 284. i)17-s + ⋯ |
| L(s) = 1 | + (−1.59 + 1.02i)2-s + (−0.0821 + 0.571i)3-s + (1.08 − 2.37i)4-s + (0.858 − 0.252i)5-s + (−0.455 − 0.998i)6-s + (−0.365 − 0.422i)7-s + (0.436 + 3.03i)8-s + (−0.319 − 0.0939i)9-s + (−1.11 + 1.28i)10-s + (0.513 + 0.329i)11-s + (1.26 + 0.815i)12-s + (−1.23 + 1.42i)13-s + (1.01 + 0.299i)14-s + (0.0735 + 0.511i)15-s + (−2.10 − 2.43i)16-s + (−0.109 − 0.238i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.809 + 0.587i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.809 + 0.587i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.0868103 - 0.267196i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0868103 - 0.267196i\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + (2.35e3 - 948. i)T \) |
| good | 2 | \( 1 + (9.04 - 5.81i)T + (13.2 - 29.1i)T^{2} \) |
| 5 | \( 1 + (-47.9 + 14.0i)T + (2.62e3 - 1.68e3i)T^{2} \) |
| 7 | \( 1 + (47.4 + 54.7i)T + (-2.39e3 + 1.66e4i)T^{2} \) |
| 11 | \( 1 + (-206. - 132. i)T + (6.69e4 + 1.46e5i)T^{2} \) |
| 13 | \( 1 + (751. - 866. i)T + (-5.28e4 - 3.67e5i)T^{2} \) |
| 17 | \( 1 + (130. + 284. i)T + (-9.29e5 + 1.07e6i)T^{2} \) |
| 19 | \( 1 + (-404. + 885. i)T + (-1.62e6 - 1.87e6i)T^{2} \) |
| 29 | \( 1 + (-2.45e3 - 5.38e3i)T + (-1.34e7 + 1.55e7i)T^{2} \) |
| 31 | \( 1 + (239. + 1.66e3i)T + (-2.74e7 + 8.06e6i)T^{2} \) |
| 37 | \( 1 + (1.45e4 + 4.28e3i)T + (5.83e7 + 3.74e7i)T^{2} \) |
| 41 | \( 1 + (676. - 198. i)T + (9.74e7 - 6.26e7i)T^{2} \) |
| 43 | \( 1 + (2.36e3 - 1.64e4i)T + (-1.41e8 - 4.14e7i)T^{2} \) |
| 47 | \( 1 + 2.88e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + (8.63e3 + 9.96e3i)T + (-5.95e7 + 4.13e8i)T^{2} \) |
| 59 | \( 1 + (2.71e4 - 3.13e4i)T + (-1.01e8 - 7.07e8i)T^{2} \) |
| 61 | \( 1 + (5.69e3 + 3.95e4i)T + (-8.10e8 + 2.37e8i)T^{2} \) |
| 67 | \( 1 + (5.90e4 - 3.79e4i)T + (5.60e8 - 1.22e9i)T^{2} \) |
| 71 | \( 1 + (3.03e4 - 1.95e4i)T + (7.49e8 - 1.64e9i)T^{2} \) |
| 73 | \( 1 + (-2.86e4 + 6.26e4i)T + (-1.35e9 - 1.56e9i)T^{2} \) |
| 79 | \( 1 + (4.26e3 - 4.91e3i)T + (-4.37e8 - 3.04e9i)T^{2} \) |
| 83 | \( 1 + (-1.29e4 - 3.79e3i)T + (3.31e9 + 2.12e9i)T^{2} \) |
| 89 | \( 1 + (1.13e4 - 7.87e4i)T + (-5.35e9 - 1.57e9i)T^{2} \) |
| 97 | \( 1 + (-4.12e4 + 1.21e4i)T + (7.22e9 - 4.64e9i)T^{2} \) |
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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
−14.66638171925490140435910894849, −13.90338486158928145222892002877, −11.76640809264257835117285991544, −10.35332069502390074625598961467, −9.546627021047107285097413687029, −9.029115035521889677404997533524, −7.32586599243218726538968331440, −6.39955155688898198067968222220, −4.96620160562910697162815038416, −1.76715543882307784421859467147,
0.20340630523303982738429357459, 1.87657935875745267314916923263, 3.01462608584255238408344434489, 6.11146040420289522359240375841, 7.54207161044465075713848264908, 8.618025388251693200719499800662, 9.871422681775403362932208687787, 10.42550674946249924435085621317, 11.96920455752518295470618115554, 12.49394694282510015065217604889
