| L(s) = 1 | + (4.86 + 6.35i)2-s + (−16.6 + 61.7i)4-s − 100. i·5-s + 277. i·7-s + (−473. + 194. i)8-s + (640. − 490. i)10-s − 1.92e3·11-s + 1.72e3i·13-s + (−1.76e3 + 1.35e3i)14-s + (−3.54e3 − 2.05e3i)16-s − 1.65e3·17-s − 9.36e3·19-s + (6.23e3 + 1.67e3i)20-s + (−9.35e3 − 1.22e4i)22-s − 1.56e4i·23-s + ⋯ |
| L(s) = 1 | + (0.608 + 0.793i)2-s + (−0.260 + 0.965i)4-s − 0.806i·5-s + 0.809i·7-s + (−0.924 + 0.380i)8-s + (0.640 − 0.490i)10-s − 1.44·11-s + 0.783i·13-s + (−0.642 + 0.492i)14-s + (−0.864 − 0.502i)16-s − 0.336·17-s − 1.36·19-s + (0.778 + 0.209i)20-s + (−0.878 − 1.14i)22-s − 1.29i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.924 + 0.380i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.924 + 0.380i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.166379 - 0.841678i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.166379 - 0.841678i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-4.86 - 6.35i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 100. iT - 1.56e4T^{2} \) |
| 7 | \( 1 - 277. iT - 1.17e5T^{2} \) |
| 11 | \( 1 + 1.92e3T + 1.77e6T^{2} \) |
| 13 | \( 1 - 1.72e3iT - 4.82e6T^{2} \) |
| 17 | \( 1 + 1.65e3T + 2.41e7T^{2} \) |
| 19 | \( 1 + 9.36e3T + 4.70e7T^{2} \) |
| 23 | \( 1 + 1.56e4iT - 1.48e8T^{2} \) |
| 29 | \( 1 - 2.84e4iT - 5.94e8T^{2} \) |
| 31 | \( 1 - 3.02e4iT - 8.87e8T^{2} \) |
| 37 | \( 1 + 7.25e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 + 3.41e4T + 4.75e9T^{2} \) |
| 43 | \( 1 + 7.12e4T + 6.32e9T^{2} \) |
| 47 | \( 1 - 1.49e5iT - 1.07e10T^{2} \) |
| 53 | \( 1 - 2.35e5iT - 2.21e10T^{2} \) |
| 59 | \( 1 - 1.35e5T + 4.21e10T^{2} \) |
| 61 | \( 1 - 1.06e5iT - 5.15e10T^{2} \) |
| 67 | \( 1 - 2.07e5T + 9.04e10T^{2} \) |
| 71 | \( 1 - 2.32e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 - 6.45e5T + 1.51e11T^{2} \) |
| 79 | \( 1 - 5.35e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 + 6.60e5T + 3.26e11T^{2} \) |
| 89 | \( 1 - 9.42e5T + 4.96e11T^{2} \) |
| 97 | \( 1 + 1.16e6T + 8.32e11T^{2} \) |
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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
−14.10104836992396260291673992783, −12.72771585755201062338576061810, −12.50515530502106408138991062504, −10.83093739257145292668049218592, −8.951579967151367060568496022221, −8.347231750332652825572621746621, −6.79167311227665776957743826260, −5.43958710079803015909218924240, −4.48433187567614449965833324639, −2.51275365198786449017612508634,
0.25052346418763311612015912875, 2.32534280478062494142888666947, 3.62208782773392680485526746560, 5.14041839483037310630197839829, 6.58924447015765786283950033995, 8.062102518616857922539162999801, 10.01213918203308138004954276870, 10.57175008295004224148256561995, 11.53833091340984455467835529644, 13.13058637232257765516057569786
