| L(s) = 1 | − 3i·3-s − 17.4i·5-s + 2.99·7-s − 9·9-s − 10.6i·11-s − 43.3i·13-s − 52.2·15-s − 37.8·17-s − 79.8i·19-s − 8.97i·21-s + 191.·23-s − 178.·25-s + 27i·27-s + 138. i·29-s − 212.·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s − 1.55i·5-s + 0.161·7-s − 0.333·9-s − 0.291i·11-s − 0.924i·13-s − 0.900·15-s − 0.540·17-s − 0.964i·19-s − 0.0932i·21-s + 1.73·23-s − 1.43·25-s + 0.192i·27-s + 0.889i·29-s − 1.22·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.301716449\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.301716449\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 3iT \) |
| good | 5 | \( 1 + 17.4iT - 125T^{2} \) |
| 7 | \( 1 - 2.99T + 343T^{2} \) |
| 11 | \( 1 + 10.6iT - 1.33e3T^{2} \) |
| 13 | \( 1 + 43.3iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 37.8T + 4.91e3T^{2} \) |
| 19 | \( 1 + 79.8iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 191.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 138. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 212.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 270. iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 441.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 64.1iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 436.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 278. iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 830. iT - 2.05e5T^{2} \) |
| 61 | \( 1 + 724. iT - 2.26e5T^{2} \) |
| 67 | \( 1 + 859. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 681.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 785.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.01e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 467. iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 510.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 234.T + 9.12e5T^{2} \) |
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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
−10.57982691571019738498804222789, −9.117204001425407658768109521329, −8.750084897933369864018800060570, −7.76216707616343718977802364088, −6.69587303980055486361878103020, −5.34368668832791519203834000922, −4.80453614088276991312085473526, −3.14988076913166215555801434426, −1.50427433505096264048914611877, −0.44473068048442218055234134422,
2.07782175514304555586530936971, 3.28186833246346149849886744446, 4.28475167253485057076311849914, 5.64715816071820087814767017211, 6.75889934025324456156843934298, 7.40211723956724102847826424364, 8.768065821015767016279995017017, 9.676492913198935647638972931862, 10.59822486991303269040299840706, 11.12831043761509208697032573424
