| L(s) = 1 | + (−1.84 + 0.760i)2-s + (1.59 − 2.54i)3-s + (2.84 − 2.81i)4-s + (−0.179 + 4.99i)5-s + (−1.01 + 5.91i)6-s + (−8.22 + 8.22i)7-s + (−3.11 + 7.36i)8-s + (−3.91 − 8.10i)9-s + (−3.46 − 9.37i)10-s + (−1.28 − 0.935i)11-s + (−2.62 − 11.7i)12-s + (2.67 − 16.8i)13-s + (8.95 − 21.4i)14-s + (12.4 + 8.42i)15-s + (0.157 − 15.9i)16-s + (−11.2 − 22.1i)17-s + ⋯ |
| L(s) = 1 | + (−0.924 + 0.380i)2-s + (0.531 − 0.847i)3-s + (0.710 − 0.703i)4-s + (−0.0359 + 0.999i)5-s + (−0.169 + 0.985i)6-s + (−1.17 + 1.17i)7-s + (−0.389 + 0.921i)8-s + (−0.435 − 0.900i)9-s + (−0.346 − 0.937i)10-s + (−0.117 − 0.0850i)11-s + (−0.218 − 0.975i)12-s + (0.205 − 1.29i)13-s + (0.639 − 1.53i)14-s + (0.827 + 0.561i)15-s + (0.00987 − 0.999i)16-s + (−0.663 − 1.30i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.795 + 0.605i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.795 + 0.605i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0924088 - 0.273813i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0924088 - 0.273813i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.84 - 0.760i)T \) |
| 3 | \( 1 + (-1.59 + 2.54i)T \) |
| 5 | \( 1 + (0.179 - 4.99i)T \) |
| good | 7 | \( 1 + (8.22 - 8.22i)T - 49iT^{2} \) |
| 11 | \( 1 + (1.28 + 0.935i)T + (37.3 + 115. i)T^{2} \) |
| 13 | \( 1 + (-2.67 + 16.8i)T + (-160. - 52.2i)T^{2} \) |
| 17 | \( 1 + (11.2 + 22.1i)T + (-169. + 233. i)T^{2} \) |
| 19 | \( 1 + (-3.50 + 10.7i)T + (-292. - 212. i)T^{2} \) |
| 23 | \( 1 + (31.9 - 5.05i)T + (503. - 163. i)T^{2} \) |
| 29 | \( 1 + (10.1 + 31.3i)T + (-680. + 494. i)T^{2} \) |
| 31 | \( 1 + (10.4 + 3.38i)T + (777. + 564. i)T^{2} \) |
| 37 | \( 1 + (-32.8 - 5.21i)T + (1.30e3 + 423. i)T^{2} \) |
| 41 | \( 1 + (-10.7 - 14.7i)T + (-519. + 1.59e3i)T^{2} \) |
| 43 | \( 1 + (-14.1 - 14.1i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (33.4 - 65.6i)T + (-1.29e3 - 1.78e3i)T^{2} \) |
| 53 | \( 1 + (9.37 - 18.3i)T + (-1.65e3 - 2.27e3i)T^{2} \) |
| 59 | \( 1 + (31.1 + 42.8i)T + (-1.07e3 + 3.31e3i)T^{2} \) |
| 61 | \( 1 + (-63.9 - 46.4i)T + (1.14e3 + 3.53e3i)T^{2} \) |
| 67 | \( 1 + (31.0 + 60.9i)T + (-2.63e3 + 3.63e3i)T^{2} \) |
| 71 | \( 1 + (26.1 + 80.5i)T + (-4.07e3 + 2.96e3i)T^{2} \) |
| 73 | \( 1 + (127. - 20.1i)T + (5.06e3 - 1.64e3i)T^{2} \) |
| 79 | \( 1 + (-32.1 - 98.9i)T + (-5.04e3 + 3.66e3i)T^{2} \) |
| 83 | \( 1 + (-32.0 - 62.9i)T + (-4.04e3 + 5.57e3i)T^{2} \) |
| 89 | \( 1 + (-2.59 - 1.88i)T + (2.44e3 + 7.53e3i)T^{2} \) |
| 97 | \( 1 + (15.6 - 30.7i)T + (-5.53e3 - 7.61e3i)T^{2} \) |
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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
−11.15224450848919049536591459372, −9.792384994056945307858102520781, −9.332473594590873520528120877219, −8.133729762396983136500087471447, −7.40034886093989084348480772098, −6.32272858686819643286462365834, −5.84766677940737081692968624425, −3.01016626350021073988507219386, −2.44494402441120583724571794836, −0.16379035995145319156723438078,
1.82735204262800496656727334254, 3.69406769627581500730565065672, 4.19265194095521544721080032812, 6.14553839321069276897142315942, 7.36753546600109632401138276243, 8.456332600283378325724609413749, 9.143689501602628885354561046753, 9.977652688276533525804157222688, 10.53962500536829963998401124313, 11.67905371336515795691550947488
