| L(s) = 1 | + (1.72 + 1.01i)2-s + (−2.92 − 0.659i)3-s + (1.95 + 3.49i)4-s + (−2.19 − 4.49i)5-s + (−4.38 − 4.09i)6-s + (−0.365 + 0.365i)7-s + (−0.159 + 7.99i)8-s + (8.12 + 3.86i)9-s + (0.753 − 9.97i)10-s + (−11.5 − 8.42i)11-s + (−3.41 − 11.5i)12-s + (2.96 − 18.7i)13-s + (−0.998 + 0.260i)14-s + (3.46 + 14.5i)15-s + (−8.36 + 13.6i)16-s + (−8.44 − 16.5i)17-s + ⋯ |
| L(s) = 1 | + (0.862 + 0.505i)2-s + (−0.975 − 0.219i)3-s + (0.488 + 0.872i)4-s + (−0.439 − 0.898i)5-s + (−0.730 − 0.683i)6-s + (−0.0521 + 0.0521i)7-s + (−0.0199 + 0.999i)8-s + (0.903 + 0.429i)9-s + (0.0753 − 0.997i)10-s + (−1.05 − 0.765i)11-s + (−0.284 − 0.958i)12-s + (0.228 − 1.44i)13-s + (−0.0713 + 0.0186i)14-s + (0.230 + 0.972i)15-s + (−0.522 + 0.852i)16-s + (−0.496 − 0.974i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.129 + 0.991i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.129 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.910782 - 0.799434i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.910782 - 0.799434i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.72 - 1.01i)T \) |
| 3 | \( 1 + (2.92 + 0.659i)T \) |
| 5 | \( 1 + (2.19 + 4.49i)T \) |
| good | 7 | \( 1 + (0.365 - 0.365i)T - 49iT^{2} \) |
| 11 | \( 1 + (11.5 + 8.42i)T + (37.3 + 115. i)T^{2} \) |
| 13 | \( 1 + (-2.96 + 18.7i)T + (-160. - 52.2i)T^{2} \) |
| 17 | \( 1 + (8.44 + 16.5i)T + (-169. + 233. i)T^{2} \) |
| 19 | \( 1 + (-3.87 + 11.9i)T + (-292. - 212. i)T^{2} \) |
| 23 | \( 1 + (-29.1 + 4.62i)T + (503. - 163. i)T^{2} \) |
| 29 | \( 1 + (9.73 + 29.9i)T + (-680. + 494. i)T^{2} \) |
| 31 | \( 1 + (56.2 + 18.2i)T + (777. + 564. i)T^{2} \) |
| 37 | \( 1 + (-66.7 - 10.5i)T + (1.30e3 + 423. i)T^{2} \) |
| 41 | \( 1 + (-9.36 - 12.8i)T + (-519. + 1.59e3i)T^{2} \) |
| 43 | \( 1 + (-5.51 - 5.51i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (3.02 - 5.92i)T + (-1.29e3 - 1.78e3i)T^{2} \) |
| 53 | \( 1 + (31.3 - 61.4i)T + (-1.65e3 - 2.27e3i)T^{2} \) |
| 59 | \( 1 + (35.4 + 48.8i)T + (-1.07e3 + 3.31e3i)T^{2} \) |
| 61 | \( 1 + (-24.0 - 17.4i)T + (1.14e3 + 3.53e3i)T^{2} \) |
| 67 | \( 1 + (41.7 + 82.0i)T + (-2.63e3 + 3.63e3i)T^{2} \) |
| 71 | \( 1 + (-6.03 - 18.5i)T + (-4.07e3 + 2.96e3i)T^{2} \) |
| 73 | \( 1 + (5.30 - 0.840i)T + (5.06e3 - 1.64e3i)T^{2} \) |
| 79 | \( 1 + (-5.46 - 16.8i)T + (-5.04e3 + 3.66e3i)T^{2} \) |
| 83 | \( 1 + (-42.1 - 82.6i)T + (-4.04e3 + 5.57e3i)T^{2} \) |
| 89 | \( 1 + (66.8 + 48.6i)T + (2.44e3 + 7.53e3i)T^{2} \) |
| 97 | \( 1 + (-56.9 + 111. i)T + (-5.53e3 - 7.61e3i)T^{2} \) |
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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
−11.20897389006285909483812750977, −11.07075351971169787941897327528, −9.301794594889387037069480725148, −7.967953335574018863944648216810, −7.44412540193271648253768952139, −5.98716674580339525624577524082, −5.31266761550986350538359359324, −4.50327260330565242364696457811, −2.91034578011434131510149006777, −0.48931401637594967446437392081,
1.89445214416010733035847383610, 3.57026576662445985509060652228, 4.51952343626627115108849770808, 5.62012102066442244973971079381, 6.71464761392319005236074039469, 7.33590272702597833012508812733, 9.318816708584412673373826256666, 10.39690412911504632119341447113, 10.94669481848250140322903075741, 11.59053899423059183848018878707
