| L(s) = 1 | + (−59.9 + 103. i)2-s + (−7.94e3 + 4.58e3i)3-s + (2.55e4 + 4.43e4i)4-s + (6.25e5 + 3.61e5i)5-s − 1.09e6i·6-s + (−1.45e6 + 5.57e6i)7-s − 1.39e7·8-s + (2.05e7 − 3.55e7i)9-s + (−7.49e7 + 4.32e7i)10-s + (6.21e7 + 1.07e8i)11-s + (−4.06e8 − 2.34e8i)12-s − 2.89e8i·13-s + (−4.91e8 − 4.85e8i)14-s − 6.62e9·15-s + (−8.37e8 + 1.45e9i)16-s + (4.99e9 − 2.88e9i)17-s + ⋯ |
| L(s) = 1 | + (−0.234 + 0.405i)2-s + (−1.21 + 0.699i)3-s + (0.390 + 0.676i)4-s + (1.60 + 0.924i)5-s − 0.654i·6-s + (−0.253 + 0.967i)7-s − 0.834·8-s + (0.477 − 0.826i)9-s + (−0.749 + 0.432i)10-s + (0.289 + 0.502i)11-s + (−0.945 − 0.545i)12-s − 0.355i·13-s + (−0.333 − 0.329i)14-s − 2.58·15-s + (−0.194 + 0.337i)16-s + (0.716 − 0.413i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 + 0.140i)\, \overline{\Lambda}(17-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s+8) \, L(s)\cr =\mathstrut & (-0.990 + 0.140i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{17}{2})\) |
\(\approx\) |
\(0.0905422 - 1.28364i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0905422 - 1.28364i\) |
| \(L(9)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (1.45e6 - 5.57e6i)T \) |
| good | 2 | \( 1 + (59.9 - 103. i)T + (-3.27e4 - 5.67e4i)T^{2} \) |
| 3 | \( 1 + (7.94e3 - 4.58e3i)T + (2.15e7 - 3.72e7i)T^{2} \) |
| 5 | \( 1 + (-6.25e5 - 3.61e5i)T + (7.62e10 + 1.32e11i)T^{2} \) |
| 11 | \( 1 + (-6.21e7 - 1.07e8i)T + (-2.29e16 + 3.97e16i)T^{2} \) |
| 13 | \( 1 + 2.89e8iT - 6.65e17T^{2} \) |
| 17 | \( 1 + (-4.99e9 + 2.88e9i)T + (2.43e19 - 4.21e19i)T^{2} \) |
| 19 | \( 1 + (7.15e9 + 4.13e9i)T + (1.44e20 + 2.49e20i)T^{2} \) |
| 23 | \( 1 + (-3.77e10 + 6.53e10i)T + (-3.06e21 - 5.31e21i)T^{2} \) |
| 29 | \( 1 - 2.99e11T + 2.50e23T^{2} \) |
| 31 | \( 1 + (9.19e11 - 5.30e11i)T + (3.63e23 - 6.29e23i)T^{2} \) |
| 37 | \( 1 + (-2.22e12 + 3.84e12i)T + (-6.16e24 - 1.06e25i)T^{2} \) |
| 41 | \( 1 - 6.89e12iT - 6.37e25T^{2} \) |
| 43 | \( 1 - 7.25e12T + 1.36e26T^{2} \) |
| 47 | \( 1 + (-1.70e13 - 9.84e12i)T + (2.83e26 + 4.91e26i)T^{2} \) |
| 53 | \( 1 + (2.75e13 + 4.76e13i)T + (-1.93e27 + 3.35e27i)T^{2} \) |
| 59 | \( 1 + (2.08e14 - 1.20e14i)T + (1.07e28 - 1.86e28i)T^{2} \) |
| 61 | \( 1 + (-2.29e14 - 1.32e14i)T + (1.83e28 + 3.18e28i)T^{2} \) |
| 67 | \( 1 + (-9.77e13 - 1.69e14i)T + (-8.24e28 + 1.42e29i)T^{2} \) |
| 71 | \( 1 - 5.19e14T + 4.16e29T^{2} \) |
| 73 | \( 1 + (-6.48e14 + 3.74e14i)T + (3.25e29 - 5.63e29i)T^{2} \) |
| 79 | \( 1 + (-3.89e14 + 6.74e14i)T + (-1.15e30 - 1.99e30i)T^{2} \) |
| 83 | \( 1 - 1.72e15iT - 5.07e30T^{2} \) |
| 89 | \( 1 + (2.22e15 + 1.28e15i)T + (7.74e30 + 1.34e31i)T^{2} \) |
| 97 | \( 1 - 3.54e15iT - 6.14e31T^{2} \) |
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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
−18.23305154638572115907418176411, −17.46017424434180602949564862388, −16.32034122304984850903235171398, −14.81657330063227962788445178211, −12.49459122620522784535234697439, −10.86693139961932902285852957981, −9.442462427119992609298854309866, −6.61650565482999424917276220540, −5.57490887766825804881199827037, −2.60721000743739306198589340174,
0.75059559490783212928674116829, 1.58130884174492098620284233096, 5.50022855752388078281928018302, 6.47327034395170705746263054977, 9.578535952288642195811601384156, 10.89191459444468590008536029924, 12.54772661001706818829309936576, 13.90391471616872760356881835846, 16.66956274701844524586438125934, 17.35405312889628218838175209689
