| L(s) = 1 | − 1.73i·3-s + 4.58·5-s − 2.99·9-s − 0.594i·11-s + 2.10·13-s − 7.94i·15-s − 1.07·17-s + 14.6i·19-s − 21.3i·23-s − 3.97·25-s + 5.19i·27-s + 15.5·29-s − 37.1i·31-s − 1.02·33-s + 33.9·37-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + 0.917·5-s − 0.333·9-s − 0.0540i·11-s + 0.161·13-s − 0.529i·15-s − 0.0630·17-s + 0.773i·19-s − 0.929i·23-s − 0.158·25-s + 0.192i·27-s + 0.534·29-s − 1.19i·31-s − 0.0311·33-s + 0.917·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.200095652\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.200095652\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 1.73iT \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 4.58T + 25T^{2} \) |
| 11 | \( 1 + 0.594iT - 121T^{2} \) |
| 13 | \( 1 - 2.10T + 169T^{2} \) |
| 17 | \( 1 + 1.07T + 289T^{2} \) |
| 19 | \( 1 - 14.6iT - 361T^{2} \) |
| 23 | \( 1 + 21.3iT - 529T^{2} \) |
| 29 | \( 1 - 15.5T + 841T^{2} \) |
| 31 | \( 1 + 37.1iT - 961T^{2} \) |
| 37 | \( 1 - 33.9T + 1.36e3T^{2} \) |
| 41 | \( 1 + 27.8T + 1.68e3T^{2} \) |
| 43 | \( 1 + 28.2iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 0.840iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 28.9T + 2.80e3T^{2} \) |
| 59 | \( 1 + 92.5iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 81.9T + 3.72e3T^{2} \) |
| 67 | \( 1 - 10.9iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 89.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 37.4T + 5.32e3T^{2} \) |
| 79 | \( 1 - 7.42iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 64.0iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 3.12T + 7.92e3T^{2} \) |
| 97 | \( 1 - 92.4T + 9.40e3T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.509727768948958193455977057667, −7.924643131212130205525638646274, −6.97018373056363701884922884711, −6.20281113588544159851415962540, −5.72243473893383734962647483720, −4.70981314299788384203623319762, −3.63820936730699563709956282936, −2.48007722872428867990885393531, −1.77931613242939719573208379435, −0.55684014476926840054658952463,
1.12649671029041723147494494105, 2.30086632155656766348098170871, 3.20175357904545609349346048460, 4.23848230285363205337688645070, 5.11439917067438083198265014375, 5.76480388482676885263663568569, 6.58213366771961119203194520664, 7.40686366480158357260770845483, 8.466171969513244761597804699883, 9.048042476104177936717502529640
