| L(s) = 1 | + (51.7 + 37.6i)2-s + (−417. + 1.28e3i)3-s + (1.26e3 + 3.89e3i)4-s + (−3.40e4 + 2.47e4i)5-s + (−6.99e4 + 5.08e4i)6-s + (9.66e4 + 2.97e5i)7-s + (−8.10e4 + 2.49e5i)8-s + (−1.88e5 − 1.36e5i)9-s − 2.69e6·10-s + (−5.59e6 − 1.77e6i)11-s − 5.53e6·12-s + (1.20e7 + 8.75e6i)13-s + (−6.18e6 + 1.90e7i)14-s + (−1.75e7 − 5.40e7i)15-s + (−1.35e7 + 9.86e6i)16-s + (−2.63e7 + 1.91e7i)17-s + ⋯ |
| L(s) = 1 | + (0.572 + 0.415i)2-s + (−0.330 + 1.01i)3-s + (0.154 + 0.475i)4-s + (−0.973 + 0.707i)5-s + (−0.612 + 0.444i)6-s + (0.310 + 0.955i)7-s + (−0.109 + 0.336i)8-s + (−0.117 − 0.0856i)9-s − 0.850·10-s + (−0.953 − 0.302i)11-s − 0.535·12-s + (0.692 + 0.503i)13-s + (−0.219 + 0.675i)14-s + (−0.397 − 1.22i)15-s + (−0.202 + 0.146i)16-s + (−0.265 + 0.192i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.563 + 0.826i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (-0.563 + 0.826i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(1.302501783\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.302501783\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-51.7 - 37.6i)T \) |
| 11 | \( 1 + (5.59e6 + 1.77e6i)T \) |
| good | 3 | \( 1 + (417. - 1.28e3i)T + (-1.28e6 - 9.37e5i)T^{2} \) |
| 5 | \( 1 + (3.40e4 - 2.47e4i)T + (3.77e8 - 1.16e9i)T^{2} \) |
| 7 | \( 1 + (-9.66e4 - 2.97e5i)T + (-7.83e10 + 5.69e10i)T^{2} \) |
| 13 | \( 1 + (-1.20e7 - 8.75e6i)T + (9.35e13 + 2.88e14i)T^{2} \) |
| 17 | \( 1 + (2.63e7 - 1.91e7i)T + (3.06e15 - 9.41e15i)T^{2} \) |
| 19 | \( 1 + (-8.48e7 + 2.61e8i)T + (-3.40e16 - 2.47e16i)T^{2} \) |
| 23 | \( 1 - 8.50e8T + 5.04e17T^{2} \) |
| 29 | \( 1 + (6.52e8 + 2.00e9i)T + (-8.30e18 + 6.03e18i)T^{2} \) |
| 31 | \( 1 + (1.22e9 + 8.88e8i)T + (7.54e18 + 2.32e19i)T^{2} \) |
| 37 | \( 1 + (6.59e9 + 2.03e10i)T + (-1.97e20 + 1.43e20i)T^{2} \) |
| 41 | \( 1 + (8.97e9 - 2.76e10i)T + (-7.48e20 - 5.43e20i)T^{2} \) |
| 43 | \( 1 + 1.35e10T + 1.71e21T^{2} \) |
| 47 | \( 1 + (2.85e10 - 8.78e10i)T + (-4.41e21 - 3.20e21i)T^{2} \) |
| 53 | \( 1 + (-2.17e11 - 1.58e11i)T + (8.04e21 + 2.47e22i)T^{2} \) |
| 59 | \( 1 + (-1.78e11 - 5.48e11i)T + (-8.49e22 + 6.17e22i)T^{2} \) |
| 61 | \( 1 + (8.56e10 - 6.21e10i)T + (5.00e22 - 1.53e23i)T^{2} \) |
| 67 | \( 1 + 5.93e11T + 5.48e23T^{2} \) |
| 71 | \( 1 + (1.20e12 - 8.75e11i)T + (3.60e23 - 1.10e24i)T^{2} \) |
| 73 | \( 1 + (5.64e10 + 1.73e11i)T + (-1.35e24 + 9.82e23i)T^{2} \) |
| 79 | \( 1 + (-7.82e11 - 5.68e11i)T + (1.44e24 + 4.43e24i)T^{2} \) |
| 83 | \( 1 + (-2.80e12 + 2.03e12i)T + (2.74e24 - 8.43e24i)T^{2} \) |
| 89 | \( 1 + 6.90e12T + 2.19e25T^{2} \) |
| 97 | \( 1 + (7.49e12 + 5.44e12i)T + (2.07e25 + 6.40e25i)T^{2} \) |
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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
−15.46956529684797864526945696242, −15.06657161802098840394459636995, −13.28314784488024100657310115975, −11.55190734452309573927613874004, −10.86093484159086975160761436627, −8.889154779269063821232173425311, −7.32366226101433934287238510776, −5.54526387704955924610680562614, −4.30423798993993801957488089725, −2.87928330113735150741865551876,
0.40320938484483237518396604146, 1.43916735338802642914735815616, 3.67098740592828938517674135239, 5.15420540315745313174888907213, 7.03617044007673425140773301676, 8.141700062582283621643616022502, 10.46240663166991365504527575834, 11.76046001278660863379892027319, 12.75195059791210163108747829148, 13.56270504968694280728881261758
