| L(s) = 1 | + (−10.1 + 6.50i)2-s + (2.21 − 15.4i)3-s + (33.5 − 73.4i)4-s + (−31.7 − 108. i)5-s + (77.9 + 170. i)6-s + (329. − 285. i)7-s + (28.7 + 199. i)8-s + (−233. − 68.4i)9-s + (1.02e3 + 889. i)10-s + (228. − 356. i)11-s + (−1.05e3 − 680. i)12-s + (−420. + 485. i)13-s + (−1.47e3 + 5.03e3i)14-s + (−1.74e3 + 250. i)15-s + (1.79e3 + 2.07e3i)16-s + (−3.04e3 + 1.39e3i)17-s + ⋯ |
| L(s) = 1 | + (−1.26 + 0.813i)2-s + (0.0821 − 0.571i)3-s + (0.524 − 1.14i)4-s + (−0.254 − 0.866i)5-s + (0.360 + 0.789i)6-s + (0.961 − 0.833i)7-s + (0.0560 + 0.390i)8-s + (−0.319 − 0.0939i)9-s + (1.02 + 0.889i)10-s + (0.171 − 0.267i)11-s + (−0.612 − 0.393i)12-s + (−0.191 + 0.220i)13-s + (−0.538 + 1.83i)14-s + (−0.515 + 0.0741i)15-s + (0.438 + 0.505i)16-s + (−0.620 + 0.283i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.703 + 0.710i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.703 + 0.710i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.216551 - 0.519269i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.216551 - 0.519269i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-2.21 + 15.4i)T \) |
| 23 | \( 1 + (6.68e3 + 1.01e4i)T \) |
| good | 2 | \( 1 + (10.1 - 6.50i)T + (26.5 - 58.2i)T^{2} \) |
| 5 | \( 1 + (31.7 + 108. i)T + (-1.31e4 + 8.44e3i)T^{2} \) |
| 7 | \( 1 + (-329. + 285. i)T + (1.67e4 - 1.16e5i)T^{2} \) |
| 11 | \( 1 + (-228. + 356. i)T + (-7.35e5 - 1.61e6i)T^{2} \) |
| 13 | \( 1 + (420. - 485. i)T + (-6.86e5 - 4.77e6i)T^{2} \) |
| 17 | \( 1 + (3.04e3 - 1.39e3i)T + (1.58e7 - 1.82e7i)T^{2} \) |
| 19 | \( 1 + (-2.94e3 - 1.34e3i)T + (3.08e7 + 3.55e7i)T^{2} \) |
| 29 | \( 1 + (6.20e3 + 1.35e4i)T + (-3.89e8 + 4.49e8i)T^{2} \) |
| 31 | \( 1 + (6.80e3 + 4.73e4i)T + (-8.51e8 + 2.50e8i)T^{2} \) |
| 37 | \( 1 + (1.43e4 - 4.87e4i)T + (-2.15e9 - 1.38e9i)T^{2} \) |
| 41 | \( 1 + (5.98e4 - 1.75e4i)T + (3.99e9 - 2.56e9i)T^{2} \) |
| 43 | \( 1 + (-3.75e4 - 5.39e3i)T + (6.06e9 + 1.78e9i)T^{2} \) |
| 47 | \( 1 - 2.33e4T + 1.07e10T^{2} \) |
| 53 | \( 1 + (2.00e5 - 1.74e5i)T + (3.15e9 - 2.19e10i)T^{2} \) |
| 59 | \( 1 + (6.97e4 - 8.04e4i)T + (-6.00e9 - 4.17e10i)T^{2} \) |
| 61 | \( 1 + (1.22e5 - 1.76e4i)T + (4.94e10 - 1.45e10i)T^{2} \) |
| 67 | \( 1 + (2.59e5 + 4.03e5i)T + (-3.75e10 + 8.22e10i)T^{2} \) |
| 71 | \( 1 + (1.41e5 - 9.07e4i)T + (5.32e10 - 1.16e11i)T^{2} \) |
| 73 | \( 1 + (1.00e5 - 2.19e5i)T + (-9.91e10 - 1.14e11i)T^{2} \) |
| 79 | \( 1 + (1.21e4 + 1.05e4i)T + (3.45e10 + 2.40e11i)T^{2} \) |
| 83 | \( 1 + (-2.20e5 + 7.50e5i)T + (-2.75e11 - 1.76e11i)T^{2} \) |
| 89 | \( 1 + (-4.09e5 - 5.89e4i)T + (4.76e11 + 1.40e11i)T^{2} \) |
| 97 | \( 1 + (2.23e5 + 7.62e5i)T + (-7.00e11 + 4.50e11i)T^{2} \) |
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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
−13.20433933137158998749425732637, −11.86980068264264039115891007751, −10.60266690788920761143413646003, −9.210553528367358412624056615636, −8.208638891430152688298816886773, −7.57744115710810849162638837391, −6.24224966653240930098018268192, −4.42583927409493230918021443730, −1.48724657263301639650178951228, −0.34696717030269991484181454512,
1.84026232656072333471469096037, 3.16318798736641305992228089500, 5.19736763301558895890299153375, 7.29754485196855449302510783406, 8.520097764582829396554807935662, 9.416427912841223507047022100217, 10.61392300930419574166310778909, 11.28909439988154020014956536996, 12.17667924041026249642190303914, 14.20732741379111817658636808180
