L(s) = 1 | + 13.1i·5-s + 4.49i·7-s − 22.3·11-s + 73.5·13-s − 42.6i·17-s + 122. i·19-s + 197.·23-s − 49.2·25-s − 14.6i·29-s + 147. i·31-s − 59.2·35-s − 234.·37-s − 396. i·41-s − 280. i·43-s + 534.·47-s + ⋯ |
L(s) = 1 | + 1.18i·5-s + 0.242i·7-s − 0.612·11-s + 1.56·13-s − 0.608i·17-s + 1.47i·19-s + 1.79·23-s − 0.393·25-s − 0.0940i·29-s + 0.852i·31-s − 0.286·35-s − 1.04·37-s − 1.50i·41-s − 0.995i·43-s + 1.65·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.339328846\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.339328846\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 - 13.1iT - 125T^{2} \) |
| 7 | \( 1 - 4.49iT - 343T^{2} \) |
| 11 | \( 1 + 22.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 73.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 42.6iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 122. iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 197.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 14.6iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 147. iT - 2.97e4T^{2} \) |
| 37 | \( 1 + 234.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 396. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 280. iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 534.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 337. iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 672.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 80.8T + 2.26e5T^{2} \) |
| 67 | \( 1 - 251. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 95.8T + 3.57e5T^{2} \) |
| 73 | \( 1 + 251.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 499. iT - 4.93e5T^{2} \) |
| 83 | \( 1 + 16.1T + 5.71e5T^{2} \) |
| 89 | \( 1 - 321. iT - 7.04e5T^{2} \) |
| 97 | \( 1 + 210.T + 9.12e5T^{2} \) |
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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
−8.941929611809071715945260242481, −8.533610059524975864116732343782, −7.31985112141733251675057864992, −6.94680441969072360624576292557, −5.88197880350636833439424955192, −5.32373618188267546949411448359, −3.89476680921070784156919186958, −3.23123245424863250638377044140, −2.31416328275322601864253895592, −1.02077146254926050778930588093,
0.62513428622253896347378804170, 1.31442067099428751059591912369, 2.71515968483111382393605968455, 3.80971720023310816502424113280, 4.72194426693474302432720263047, 5.35979912587864741732438652835, 6.31783484152370809160023670561, 7.19847755557267378806239180674, 8.182225868969042637941060553292, 8.807775001488240923012312829109