| L(s) = 1 | + (−12.2 + 7.88i)2-s + (−2.21 + 15.4i)3-s + (61.6 − 135. i)4-s + (34.1 + 116. i)5-s + (−94.3 − 206. i)6-s + (289. − 251. i)7-s + (175. + 1.21e3i)8-s + (−233. − 68.4i)9-s + (−1.33e3 − 1.15e3i)10-s + (−378. + 589. i)11-s + (1.94e3 + 1.25e3i)12-s + (−1.57e3 + 1.81e3i)13-s + (−1.57e3 + 5.36e3i)14-s + (−1.86e3 + 268. i)15-s + (−5.52e3 − 6.38e3i)16-s + (−6.08e3 + 2.77e3i)17-s + ⋯ |
| L(s) = 1 | + (−1.53 + 0.985i)2-s + (−0.0821 + 0.571i)3-s + (0.963 − 2.11i)4-s + (0.273 + 0.929i)5-s + (−0.436 − 0.956i)6-s + (0.844 − 0.731i)7-s + (0.342 + 2.37i)8-s + (−0.319 − 0.0939i)9-s + (−1.33 − 1.15i)10-s + (−0.284 + 0.442i)11-s + (1.12 + 0.724i)12-s + (−0.715 + 0.825i)13-s + (−0.573 + 1.95i)14-s + (−0.553 + 0.0796i)15-s + (−1.34 − 1.55i)16-s + (−1.23 + 0.565i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.155 + 0.987i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.155 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.128016 - 0.149694i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.128016 - 0.149694i\) |
| \(L(4)\) |
|
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 | \( 1 + (-1.13e4 - 4.33e3i)T \) |
| good | 2 | \( 1 + (12.2 - 7.88i)T + (26.5 - 58.2i)T^{2} \) |
| 5 | \( 1 + (-34.1 - 116. i)T + (-1.31e4 + 8.44e3i)T^{2} \) |
| 7 | \( 1 + (-289. + 251. i)T + (1.67e4 - 1.16e5i)T^{2} \) |
| 11 | \( 1 + (378. - 589. i)T + (-7.35e5 - 1.61e6i)T^{2} \) |
| 13 | \( 1 + (1.57e3 - 1.81e3i)T + (-6.86e5 - 4.77e6i)T^{2} \) |
| 17 | \( 1 + (6.08e3 - 2.77e3i)T + (1.58e7 - 1.82e7i)T^{2} \) |
| 19 | \( 1 + (4.43e3 + 2.02e3i)T + (3.08e7 + 3.55e7i)T^{2} \) |
| 29 | \( 1 + (1.57e4 + 3.44e4i)T + (-3.89e8 + 4.49e8i)T^{2} \) |
| 31 | \( 1 + (-1.34e3 - 9.36e3i)T + (-8.51e8 + 2.50e8i)T^{2} \) |
| 37 | \( 1 + (5.31e3 - 1.81e4i)T + (-2.15e9 - 1.38e9i)T^{2} \) |
| 41 | \( 1 + (1.05e5 - 3.09e4i)T + (3.99e9 - 2.56e9i)T^{2} \) |
| 43 | \( 1 + (6.19e4 + 8.90e3i)T + (6.06e9 + 1.78e9i)T^{2} \) |
| 47 | \( 1 - 6.04e4T + 1.07e10T^{2} \) |
| 53 | \( 1 + (-6.88e4 + 5.96e4i)T + (3.15e9 - 2.19e10i)T^{2} \) |
| 59 | \( 1 + (-1.96e5 + 2.26e5i)T + (-6.00e9 - 4.17e10i)T^{2} \) |
| 61 | \( 1 + (3.41e5 - 4.90e4i)T + (4.94e10 - 1.45e10i)T^{2} \) |
| 67 | \( 1 + (2.30e5 + 3.57e5i)T + (-3.75e10 + 8.22e10i)T^{2} \) |
| 71 | \( 1 + (3.89e5 - 2.50e5i)T + (5.32e10 - 1.16e11i)T^{2} \) |
| 73 | \( 1 + (-2.66e5 + 5.84e5i)T + (-9.91e10 - 1.14e11i)T^{2} \) |
| 79 | \( 1 + (-2.13e5 - 1.85e5i)T + (3.45e10 + 2.40e11i)T^{2} \) |
| 83 | \( 1 + (1.34e5 - 4.59e5i)T + (-2.75e11 - 1.76e11i)T^{2} \) |
| 89 | \( 1 + (1.17e6 + 1.68e5i)T + (4.76e11 + 1.40e11i)T^{2} \) |
| 97 | \( 1 + (-7.76e4 - 2.64e5i)T + (-7.00e11 + 4.50e11i)T^{2} \) |
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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
−15.02861042885620837106687135151, −13.76713362561283375748012960637, −11.29555801354569302201033013013, −10.60035901541624526594807216983, −9.724390584205791768836344967617, −8.542491901438466038431480249518, −7.28559913898566224082701811850, −6.48200732527131792928269128996, −4.71735423153845987782527526016, −1.97156535423401802386120989411,
0.12772483388035693823483335275, 1.47412813765739897018149122424, 2.62892599235300969078682013498, 5.18183004837847236474768848945, 7.27003649352976896493558376637, 8.587130685680389909416369030865, 8.895647013083218572377467248379, 10.47699565811316962293197844453, 11.45253514154420192091737974665, 12.41354564412253717774460873012
