| L(s) = 1 | + (0.991 + 1.14i)2-s + (13.1 + 8.42i)3-s + (8.78 − 61.0i)4-s + (−45.4 − 20.7i)5-s + (3.35 + 23.3i)6-s + (28.4 − 96.9i)7-s + (160. − 102. i)8-s + (100. + 221. i)9-s + (−21.3 − 72.5i)10-s + (−1.34e3 − 1.16e3i)11-s + (629. − 726. i)12-s + (−2.56e3 + 753. i)13-s + (139. − 63.5i)14-s + (−420. − 654. i)15-s + (−3.51e3 − 1.03e3i)16-s + (2.49e3 − 359. i)17-s + ⋯ |
| L(s) = 1 | + (0.123 + 0.143i)2-s + (0.485 + 0.312i)3-s + (0.137 − 0.954i)4-s + (−0.363 − 0.165i)5-s + (0.0155 + 0.108i)6-s + (0.0829 − 0.282i)7-s + (0.312 − 0.201i)8-s + (0.138 + 0.303i)9-s + (−0.0213 − 0.0725i)10-s + (−1.00 − 0.872i)11-s + (0.364 − 0.420i)12-s + (−1.16 + 0.342i)13-s + (0.0507 − 0.0231i)14-s + (−0.124 − 0.194i)15-s + (−0.857 − 0.251i)16-s + (0.508 − 0.0731i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.502 + 0.864i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.502 + 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.688956 - 1.19687i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.688956 - 1.19687i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-13.1 - 8.42i)T \) |
| 23 | \( 1 + (-4.83e3 + 1.11e4i)T \) |
| good | 2 | \( 1 + (-0.991 - 1.14i)T + (-9.10 + 63.3i)T^{2} \) |
| 5 | \( 1 + (45.4 + 20.7i)T + (1.02e4 + 1.18e4i)T^{2} \) |
| 7 | \( 1 + (-28.4 + 96.9i)T + (-9.89e4 - 6.36e4i)T^{2} \) |
| 11 | \( 1 + (1.34e3 + 1.16e3i)T + (2.52e5 + 1.75e6i)T^{2} \) |
| 13 | \( 1 + (2.56e3 - 753. i)T + (4.06e6 - 2.60e6i)T^{2} \) |
| 17 | \( 1 + (-2.49e3 + 359. i)T + (2.31e7 - 6.80e6i)T^{2} \) |
| 19 | \( 1 + (187. + 27.0i)T + (4.51e7 + 1.32e7i)T^{2} \) |
| 29 | \( 1 + (3.40e3 + 2.36e4i)T + (-5.70e8 + 1.67e8i)T^{2} \) |
| 31 | \( 1 + (-1.55e4 + 1.00e4i)T + (3.68e8 - 8.07e8i)T^{2} \) |
| 37 | \( 1 + (7.60e4 - 3.47e4i)T + (1.68e9 - 1.93e9i)T^{2} \) |
| 41 | \( 1 + (1.16e4 - 2.55e4i)T + (-3.11e9 - 3.58e9i)T^{2} \) |
| 43 | \( 1 + (-4.81e4 + 7.48e4i)T + (-2.62e9 - 5.75e9i)T^{2} \) |
| 47 | \( 1 - 1.45e5T + 1.07e10T^{2} \) |
| 53 | \( 1 + (-3.84e4 + 1.30e5i)T + (-1.86e10 - 1.19e10i)T^{2} \) |
| 59 | \( 1 + (1.29e5 - 3.81e4i)T + (3.54e10 - 2.28e10i)T^{2} \) |
| 61 | \( 1 + (-1.34e5 - 2.08e5i)T + (-2.14e10 + 4.68e10i)T^{2} \) |
| 67 | \( 1 + (1.41e5 - 1.22e5i)T + (1.28e10 - 8.95e10i)T^{2} \) |
| 71 | \( 1 + (-1.48e5 - 1.71e5i)T + (-1.82e10 + 1.26e11i)T^{2} \) |
| 73 | \( 1 + (8.78e4 - 6.11e5i)T + (-1.45e11 - 4.26e10i)T^{2} \) |
| 79 | \( 1 + (-1.98e4 - 6.75e4i)T + (-2.04e11 + 1.31e11i)T^{2} \) |
| 83 | \( 1 + (-2.57e5 + 1.17e5i)T + (2.14e11 - 2.47e11i)T^{2} \) |
| 89 | \( 1 + (-5.02e5 + 7.81e5i)T + (-2.06e11 - 4.52e11i)T^{2} \) |
| 97 | \( 1 + (-7.95e5 - 3.63e5i)T + (5.45e11 + 6.29e11i)T^{2} \) |
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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
−13.50953974412318317648395139460, −11.98633682125892233166643011749, −10.64064846871025887268706391561, −9.888330542424715291898191011921, −8.468115649853898862941279301301, −7.22148737827529519368295963288, −5.59988690335506571051145244812, −4.39611646040295486505488264609, −2.49337514793624704723144339300, −0.46134877959933340240945183494,
2.21712567625355118540681054680, 3.40478142980042954250001456150, 5.06500898346004523186790166742, 7.29984310123391259465598953968, 7.72658881559557610102040967472, 9.153859075344830144460238506315, 10.58664378839941281183421664992, 12.08005261644187099038541473680, 12.57897512786265644693546672085, 13.71540214343395445884115091448
