L(s) = 1 | + (−1.92 + 2.07i)2-s + (2.12 − 2.12i)3-s + (−0.570 − 7.97i)4-s + (−7.29 − 7.29i)5-s + (0.302 + 8.47i)6-s − 22.1i·7-s + (17.6 + 14.1i)8-s − 8.99i·9-s + (29.1 − 1.04i)10-s + (8.24 + 8.24i)11-s + (−18.1 − 15.7i)12-s + (51.9 − 51.9i)13-s + (45.8 + 42.7i)14-s − 30.9·15-s + (−63.3 + 9.10i)16-s − 58.7·17-s + ⋯ |
L(s) = 1 | + (−0.681 + 0.731i)2-s + (0.408 − 0.408i)3-s + (−0.0713 − 0.997i)4-s + (−0.652 − 0.652i)5-s + (0.0205 + 0.576i)6-s − 1.19i·7-s + (0.778 + 0.627i)8-s − 0.333i·9-s + (0.921 − 0.0329i)10-s + (0.225 + 0.225i)11-s + (−0.436 − 0.378i)12-s + (1.10 − 1.10i)13-s + (0.875 + 0.815i)14-s − 0.532·15-s + (−0.989 + 0.142i)16-s − 0.837·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.510 + 0.860i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.510 + 0.860i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.798355 - 0.454654i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.798355 - 0.454654i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.92 - 2.07i)T \) |
| 3 | \( 1 + (-2.12 + 2.12i)T \) |
good | 5 | \( 1 + (7.29 + 7.29i)T + 125iT^{2} \) |
| 7 | \( 1 + 22.1iT - 343T^{2} \) |
| 11 | \( 1 + (-8.24 - 8.24i)T + 1.33e3iT^{2} \) |
| 13 | \( 1 + (-51.9 + 51.9i)T - 2.19e3iT^{2} \) |
| 17 | \( 1 + 58.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + (54.5 - 54.5i)T - 6.85e3iT^{2} \) |
| 23 | \( 1 - 117. iT - 1.21e4T^{2} \) |
| 29 | \( 1 + (-175. + 175. i)T - 2.43e4iT^{2} \) |
| 31 | \( 1 + 6.58T + 2.97e4T^{2} \) |
| 37 | \( 1 + (-265. - 265. i)T + 5.06e4iT^{2} \) |
| 41 | \( 1 + 98.7iT - 6.89e4T^{2} \) |
| 43 | \( 1 + (-347. - 347. i)T + 7.95e4iT^{2} \) |
| 47 | \( 1 + 141.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (210. + 210. i)T + 1.48e5iT^{2} \) |
| 59 | \( 1 + (-427. - 427. i)T + 2.05e5iT^{2} \) |
| 61 | \( 1 + (-178. + 178. i)T - 2.26e5iT^{2} \) |
| 67 | \( 1 + (-480. + 480. i)T - 3.00e5iT^{2} \) |
| 71 | \( 1 + 884. iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 794. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 421.T + 4.93e5T^{2} \) |
| 83 | \( 1 + (167. - 167. i)T - 5.71e5iT^{2} \) |
| 89 | \( 1 + 664. iT - 7.04e5T^{2} \) |
| 97 | \( 1 + 1.08e3T + 9.12e5T^{2} \) |
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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
−15.24568613772952653522484165332, −13.89538868537061058436654150981, −12.99177879254855796553550566548, −11.22839395012379550452489251251, −9.963994049277243860455130653173, −8.428419385939814932167284111149, −7.75694532232247241357358963237, −6.32047030299911131094372470059, −4.25361937003264696669497999183, −0.911332021036830210487955812288,
2.53329918797176210966640132807, 4.10489259130613309055192573450, 6.74146067920465247403293775318, 8.532756112205202863181465672666, 9.090029451013187823936069576703, 10.81424442384918648557130138513, 11.48190407364180775075747979479, 12.78506909337297744992985601191, 14.28136355898355303924534776898, 15.55590817692512096961694541241