| L(s) = 1 | + (0.578 − 1.91i)2-s + (2.99 − 0.0861i)3-s + (−3.33 − 2.21i)4-s + (2.66 − 4.23i)5-s + (1.57 − 5.79i)6-s + (6.36 − 6.36i)7-s + (−6.16 + 5.09i)8-s + (8.98 − 0.516i)9-s + (−6.55 − 7.54i)10-s + (−1.78 + 5.50i)11-s + (−10.1 − 6.35i)12-s + (4.26 + 2.17i)13-s + (−8.50 − 15.8i)14-s + (7.62 − 12.9i)15-s + (6.18 + 14.7i)16-s + (−2.00 + 12.6i)17-s + ⋯ |
| L(s) = 1 | + (0.289 − 0.957i)2-s + (0.999 − 0.0287i)3-s + (−0.832 − 0.553i)4-s + (0.532 − 0.846i)5-s + (0.261 − 0.965i)6-s + (0.909 − 0.909i)7-s + (−0.770 + 0.636i)8-s + (0.998 − 0.0574i)9-s + (−0.655 − 0.754i)10-s + (−0.162 + 0.500i)11-s + (−0.848 − 0.529i)12-s + (0.327 + 0.167i)13-s + (−0.607 − 1.13i)14-s + (0.508 − 0.861i)15-s + (0.386 + 0.922i)16-s + (−0.117 + 0.742i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.469 + 0.883i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.469 + 0.883i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.42380 - 2.36823i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.42380 - 2.36823i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.578 + 1.91i)T \) |
| 3 | \( 1 + (-2.99 + 0.0861i)T \) |
| 5 | \( 1 + (-2.66 + 4.23i)T \) |
| good | 7 | \( 1 + (-6.36 + 6.36i)T - 49iT^{2} \) |
| 11 | \( 1 + (1.78 - 5.50i)T + (-97.8 - 71.1i)T^{2} \) |
| 13 | \( 1 + (-4.26 - 2.17i)T + (99.3 + 136. i)T^{2} \) |
| 17 | \( 1 + (2.00 - 12.6i)T + (-274. - 89.3i)T^{2} \) |
| 19 | \( 1 + (18.1 - 13.1i)T + (111. - 343. i)T^{2} \) |
| 23 | \( 1 + (6.61 + 12.9i)T + (-310. + 427. i)T^{2} \) |
| 29 | \( 1 + (37.6 + 27.3i)T + (259. + 799. i)T^{2} \) |
| 31 | \( 1 + (-19.3 - 26.6i)T + (-296. + 913. i)T^{2} \) |
| 37 | \( 1 + (-29.8 + 58.5i)T + (-804. - 1.10e3i)T^{2} \) |
| 41 | \( 1 + (22.2 - 7.22i)T + (1.35e3 - 988. i)T^{2} \) |
| 43 | \( 1 + (-43.9 - 43.9i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (-11.1 - 70.1i)T + (-2.10e3 + 682. i)T^{2} \) |
| 53 | \( 1 + (3.01 + 19.0i)T + (-2.67e3 + 868. i)T^{2} \) |
| 59 | \( 1 + (23.0 - 7.49i)T + (2.81e3 - 2.04e3i)T^{2} \) |
| 61 | \( 1 + (20.7 - 63.8i)T + (-3.01e3 - 2.18e3i)T^{2} \) |
| 67 | \( 1 + (0.781 - 4.93i)T + (-4.26e3 - 1.38e3i)T^{2} \) |
| 71 | \( 1 + (36.7 + 26.6i)T + (1.55e3 + 4.79e3i)T^{2} \) |
| 73 | \( 1 + (-25.1 - 49.4i)T + (-3.13e3 + 4.31e3i)T^{2} \) |
| 79 | \( 1 + (-76.7 - 55.7i)T + (1.92e3 + 5.93e3i)T^{2} \) |
| 83 | \( 1 + (15.8 - 100. i)T + (-6.55e3 - 2.12e3i)T^{2} \) |
| 89 | \( 1 + (-21.0 + 64.7i)T + (-6.40e3 - 4.65e3i)T^{2} \) |
| 97 | \( 1 + (24.1 + 152. i)T + (-8.94e3 + 2.90e3i)T^{2} \) |
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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
−11.06137399308096181613360440326, −10.28472243291181096456003185431, −9.438372104828023988756275429685, −8.505640950486908158632680168706, −7.76995632370533787939017480811, −6.01874582335639859975033230935, −4.48858155143052102478962674962, −4.07678015245549933534933865255, −2.22592236571135873827719451213, −1.30825850843835995262088191983,
2.26384655043857798884620717632, 3.43669518692624621847189583721, 4.87743848147816121758043444809, 5.95168121347319733488346062174, 7.04680678807764162320698616091, 7.990034506892322147225434079289, 8.804108634178120191572614885848, 9.547476369615606927573536450585, 10.82933206616590723094230691395, 11.95055371465915148427430954690
