| L(s) = 1 | + (−596. − 418. i)3-s − 2.39e4i·5-s − 3.21e4·7-s + (1.80e5 + 4.99e5i)9-s + 2.66e6i·11-s + 7.50e6·13-s + (−1.00e7 + 1.42e7i)15-s − 1.32e6i·17-s − 3.25e7·19-s + (1.91e7 + 1.34e7i)21-s + 8.48e7i·23-s − 3.27e8·25-s + (1.01e8 − 3.73e8i)27-s + 8.40e8i·29-s + 1.20e9·31-s + ⋯ |
| L(s) = 1 | + (−0.818 − 0.574i)3-s − 1.53i·5-s − 0.273·7-s + (0.340 + 0.940i)9-s + 1.50i·11-s + 1.55·13-s + (−0.878 + 1.25i)15-s − 0.0550i·17-s − 0.692·19-s + (0.223 + 0.156i)21-s + 0.573i·23-s − 1.34·25-s + (0.261 − 0.965i)27-s + 1.41i·29-s + 1.35·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.818 + 0.574i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (0.818 + 0.574i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{13}{2})\) |
\(\approx\) |
\(1.455542112\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.455542112\) |
| \(L(7)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (596. + 418. i)T \) |
| good | 5 | \( 1 + 2.39e4iT - 2.44e8T^{2} \) |
| 7 | \( 1 + 3.21e4T + 1.38e10T^{2} \) |
| 11 | \( 1 - 2.66e6iT - 3.13e12T^{2} \) |
| 13 | \( 1 - 7.50e6T + 2.32e13T^{2} \) |
| 17 | \( 1 + 1.32e6iT - 5.82e14T^{2} \) |
| 19 | \( 1 + 3.25e7T + 2.21e15T^{2} \) |
| 23 | \( 1 - 8.48e7iT - 2.19e16T^{2} \) |
| 29 | \( 1 - 8.40e8iT - 3.53e17T^{2} \) |
| 31 | \( 1 - 1.20e9T + 7.87e17T^{2} \) |
| 37 | \( 1 - 1.09e9T + 6.58e18T^{2} \) |
| 41 | \( 1 + 5.50e9iT - 2.25e19T^{2} \) |
| 43 | \( 1 - 6.27e9T + 3.99e19T^{2} \) |
| 47 | \( 1 + 4.76e9iT - 1.16e20T^{2} \) |
| 53 | \( 1 - 2.32e10iT - 4.91e20T^{2} \) |
| 59 | \( 1 + 1.03e10iT - 1.77e21T^{2} \) |
| 61 | \( 1 - 4.64e10T + 2.65e21T^{2} \) |
| 67 | \( 1 + 3.54e10T + 8.18e21T^{2} \) |
| 71 | \( 1 + 2.45e11iT - 1.64e22T^{2} \) |
| 73 | \( 1 - 2.38e11T + 2.29e22T^{2} \) |
| 79 | \( 1 - 3.79e10T + 5.90e22T^{2} \) |
| 83 | \( 1 - 2.26e11iT - 1.06e23T^{2} \) |
| 89 | \( 1 - 8.22e11iT - 2.46e23T^{2} \) |
| 97 | \( 1 - 3.60e11T + 6.93e23T^{2} \) |
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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
−12.73387777405820445037126362609, −12.12730811011925451016328324434, −10.71591139338993151393666534140, −9.272473240966653346226450619976, −8.063130305240283859193427041712, −6.61955648931601434033083746399, −5.32844297997447631091314304053, −4.26951689546699708629594880087, −1.76241915453433969668043525945, −0.825986681738038834915303966092,
0.67834108436230164139687359244, 2.94051396956640476046244315843, 3.98780048864468071739503091134, 6.07582405843986021826280185439, 6.42738316366795149689091955584, 8.359549683156933934758577715274, 9.988948390442688207965127183514, 10.98428360714476329738660574261, 11.43421665285301068503134129249, 13.25756010686103025362988370222
