| L(s) = 1 | + (1.46 − 5.46i)2-s + (6.55 + 11.3i)3-s + (−27.7 − 16i)4-s + (−97.4 − 97.4i)5-s + (71.6 − 19.2i)6-s + (−87.6 − 327. i)7-s + (−128 + 127. i)8-s + (278. − 482. i)9-s + (−675. + 389. i)10-s + (−873. − 233. i)11-s − 419. i·12-s + (−2.06e3 + 756. i)13-s − 1.91e3·14-s + (467. − 1.74e3i)15-s + (511. + 886. i)16-s + (2.11e3 + 1.21e3i)17-s + ⋯ |
| L(s) = 1 | + (0.183 − 0.683i)2-s + (0.242 + 0.420i)3-s + (−0.433 − 0.250i)4-s + (−0.779 − 0.779i)5-s + (0.331 − 0.0888i)6-s + (−0.255 − 0.954i)7-s + (−0.250 + 0.249i)8-s + (0.382 − 0.661i)9-s + (−0.675 + 0.389i)10-s + (−0.656 − 0.175i)11-s − 0.242i·12-s + (−0.938 + 0.344i)13-s − 0.698·14-s + (0.138 − 0.517i)15-s + (0.124 + 0.216i)16-s + (0.429 + 0.248i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.794 + 0.606i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.794 + 0.606i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.382086 - 1.13010i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.382086 - 1.13010i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.46 + 5.46i)T \) |
| 13 | \( 1 + (2.06e3 - 756. i)T \) |
| good | 3 | \( 1 + (-6.55 - 11.3i)T + (-364.5 + 631. i)T^{2} \) |
| 5 | \( 1 + (97.4 + 97.4i)T + 1.56e4iT^{2} \) |
| 7 | \( 1 + (87.6 + 327. i)T + (-1.01e5 + 5.88e4i)T^{2} \) |
| 11 | \( 1 + (873. + 233. i)T + (1.53e6 + 8.85e5i)T^{2} \) |
| 17 | \( 1 + (-2.11e3 - 1.21e3i)T + (1.20e7 + 2.09e7i)T^{2} \) |
| 19 | \( 1 + (-1.03e4 + 2.78e3i)T + (4.07e7 - 2.35e7i)T^{2} \) |
| 23 | \( 1 + (-2.52e3 + 1.45e3i)T + (7.40e7 - 1.28e8i)T^{2} \) |
| 29 | \( 1 + (-1.53e4 - 2.65e4i)T + (-2.97e8 + 5.15e8i)T^{2} \) |
| 31 | \( 1 + (2.83e4 + 2.83e4i)T + 8.87e8iT^{2} \) |
| 37 | \( 1 + (-8.58e3 - 2.29e3i)T + (2.22e9 + 1.28e9i)T^{2} \) |
| 41 | \( 1 + (-2.41e4 + 9.01e4i)T + (-4.11e9 - 2.37e9i)T^{2} \) |
| 43 | \( 1 + (4.21e4 + 2.43e4i)T + (3.16e9 + 5.47e9i)T^{2} \) |
| 47 | \( 1 + (9.19e4 - 9.19e4i)T - 1.07e10iT^{2} \) |
| 53 | \( 1 - 6.28e4T + 2.21e10T^{2} \) |
| 59 | \( 1 + (-2.96e4 - 1.10e5i)T + (-3.65e10 + 2.10e10i)T^{2} \) |
| 61 | \( 1 + (5.09e3 - 8.83e3i)T + (-2.57e10 - 4.46e10i)T^{2} \) |
| 67 | \( 1 + (-1.32e5 + 4.94e5i)T + (-7.83e10 - 4.52e10i)T^{2} \) |
| 71 | \( 1 + (4.36e5 - 1.16e5i)T + (1.10e11 - 6.40e10i)T^{2} \) |
| 73 | \( 1 + (-4.24e5 + 4.24e5i)T - 1.51e11iT^{2} \) |
| 79 | \( 1 - 8.14e4T + 2.43e11T^{2} \) |
| 83 | \( 1 + (3.20e5 + 3.20e5i)T + 3.26e11iT^{2} \) |
| 89 | \( 1 + (1.36e5 + 3.65e4i)T + (4.30e11 + 2.48e11i)T^{2} \) |
| 97 | \( 1 + (-1.64e6 + 4.39e5i)T + (7.21e11 - 4.16e11i)T^{2} \) |
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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
−15.69092968702429221975057217796, −14.29671445533748637698256604915, −12.91105925608571170925188899570, −11.89249854462833694250007790654, −10.34202436867669293780128712497, −9.186089907691400653437665995508, −7.46615022108455707167742515422, −4.81356710158111932996683388908, −3.50881982478658571142264060888, −0.65035779205494306550095001399,
2.91336550756125164976833991539, 5.25029189831059101897107275252, 7.16994912548619204544968028445, 8.008154576909949141870430492391, 9.919075165615057247916234599565, 11.76555549737460721341720591492, 12.95883663195173446222843783183, 14.36943535157164075407088172044, 15.39972584412325664406631242678, 16.26182944984284050849617794636
