| L(s) = 1 | + (8.66 − 5.56i)2-s + (−2.21 + 15.4i)3-s + (17.4 − 38.2i)4-s + (−44.7 − 152. i)5-s + (66.6 + 145. i)6-s + (−478. + 414. i)7-s + (32.2 + 224. i)8-s + (−233. − 68.4i)9-s + (−1.23e3 − 1.07e3i)10-s + (−466. + 726. i)11-s + (550. + 353. i)12-s + (569. − 656. i)13-s + (−1.83e3 + 6.25e3i)14-s + (2.44e3 − 352. i)15-s + (3.28e3 + 3.79e3i)16-s + (−3.17e3 + 1.45e3i)17-s + ⋯ |
| L(s) = 1 | + (1.08 − 0.695i)2-s + (−0.0821 + 0.571i)3-s + (0.272 − 0.596i)4-s + (−0.357 − 1.21i)5-s + (0.308 + 0.675i)6-s + (−1.39 + 1.20i)7-s + (0.0629 + 0.437i)8-s + (−0.319 − 0.0939i)9-s + (−1.23 − 1.07i)10-s + (−0.350 + 0.545i)11-s + (0.318 + 0.204i)12-s + (0.259 − 0.298i)13-s + (−0.668 + 2.27i)14-s + (0.725 − 0.104i)15-s + (0.802 + 0.926i)16-s + (−0.646 + 0.295i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.550 - 0.834i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.550 - 0.834i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.359675 + 0.667859i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.359675 + 0.667859i\) |
| \(L(4)\) |
|
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 | \( 1 + (1.13e4 - 4.45e3i)T \) |
| good | 2 | \( 1 + (-8.66 + 5.56i)T + (26.5 - 58.2i)T^{2} \) |
| 5 | \( 1 + (44.7 + 152. i)T + (-1.31e4 + 8.44e3i)T^{2} \) |
| 7 | \( 1 + (478. - 414. i)T + (1.67e4 - 1.16e5i)T^{2} \) |
| 11 | \( 1 + (466. - 726. i)T + (-7.35e5 - 1.61e6i)T^{2} \) |
| 13 | \( 1 + (-569. + 656. i)T + (-6.86e5 - 4.77e6i)T^{2} \) |
| 17 | \( 1 + (3.17e3 - 1.45e3i)T + (1.58e7 - 1.82e7i)T^{2} \) |
| 19 | \( 1 + (6.76e3 + 3.08e3i)T + (3.08e7 + 3.55e7i)T^{2} \) |
| 29 | \( 1 + (-8.33e3 - 1.82e4i)T + (-3.89e8 + 4.49e8i)T^{2} \) |
| 31 | \( 1 + (-234. - 1.62e3i)T + (-8.51e8 + 2.50e8i)T^{2} \) |
| 37 | \( 1 + (-2.13e4 + 7.26e4i)T + (-2.15e9 - 1.38e9i)T^{2} \) |
| 41 | \( 1 + (4.42e3 - 1.29e3i)T + (3.99e9 - 2.56e9i)T^{2} \) |
| 43 | \( 1 + (1.32e3 + 190. i)T + (6.06e9 + 1.78e9i)T^{2} \) |
| 47 | \( 1 + 1.24e4T + 1.07e10T^{2} \) |
| 53 | \( 1 + (-3.53e4 + 3.06e4i)T + (3.15e9 - 2.19e10i)T^{2} \) |
| 59 | \( 1 + (1.81e5 - 2.09e5i)T + (-6.00e9 - 4.17e10i)T^{2} \) |
| 61 | \( 1 + (3.81e5 - 5.48e4i)T + (4.94e10 - 1.45e10i)T^{2} \) |
| 67 | \( 1 + (-1.15e5 - 1.80e5i)T + (-3.75e10 + 8.22e10i)T^{2} \) |
| 71 | \( 1 + (-5.38e5 + 3.46e5i)T + (5.32e10 - 1.16e11i)T^{2} \) |
| 73 | \( 1 + (-6.41e4 + 1.40e5i)T + (-9.91e10 - 1.14e11i)T^{2} \) |
| 79 | \( 1 + (-5.27e5 - 4.57e5i)T + (3.45e10 + 2.40e11i)T^{2} \) |
| 83 | \( 1 + (1.28e5 - 4.37e5i)T + (-2.75e11 - 1.76e11i)T^{2} \) |
| 89 | \( 1 + (-6.14e4 - 8.83e3i)T + (4.76e11 + 1.40e11i)T^{2} \) |
| 97 | \( 1 + (-2.37e4 - 8.09e4i)T + (-7.00e11 + 4.50e11i)T^{2} \) |
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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
−13.43343412025105684983758458444, −12.54528827055366385750557199491, −12.24756846969257808012307783798, −10.78577761581991929137578767979, −9.350845244597190130314675003711, −8.452529826193743492220729016110, −6.04490417026597194594855849584, −4.95843141779785518033404141154, −3.82931640496172951703396784803, −2.40707204257180977468021246567,
0.19701104276067973806137364258, 3.06822769713822837137057534755, 4.14693342702546311150148554385, 6.36763896698108526375999805832, 6.56414527120061135736660529828, 7.77751604563556276331473006013, 9.971218710381275003301348448848, 10.95642815007750094546298424455, 12.47745185841783569285647063902, 13.54166317522620265568645182704
