| L(s) = 1 | + (−20.0 + 5.37i)2-s + (16.6 − 62.0i)3-s + (262. − 151. i)4-s + (227. + 162. i)5-s + 1.33e3i·6-s + (431. − 798. i)7-s + (−2.57e3 + 2.57e3i)8-s + (−1.67e3 − 966. i)9-s + (−5.43e3 − 2.02e3i)10-s + (3.09e3 + 5.36e3i)11-s + (−5.03e3 − 1.88e4i)12-s + (1.82e3 + 1.82e3i)13-s + (−4.36e3 + 1.83e4i)14-s + (1.38e4 − 1.14e4i)15-s + (1.83e4 − 3.18e4i)16-s + (3.11e4 + 8.34e3i)17-s + ⋯ |
| L(s) = 1 | + (−1.77 + 0.475i)2-s + (0.355 − 1.32i)3-s + (2.05 − 1.18i)4-s + (0.814 + 0.580i)5-s + 2.51i·6-s + (0.475 − 0.879i)7-s + (−1.77 + 1.77i)8-s + (−0.765 − 0.441i)9-s + (−1.71 − 0.641i)10-s + (0.701 + 1.21i)11-s + (−0.841 − 3.14i)12-s + (0.230 + 0.230i)13-s + (−0.424 + 1.78i)14-s + (1.05 − 0.873i)15-s + (1.12 − 1.94i)16-s + (1.53 + 0.411i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.772 + 0.634i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.772 + 0.634i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.06904 - 0.382802i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.06904 - 0.382802i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-227. - 162. i)T \) |
| 7 | \( 1 + (-431. + 798. i)T \) |
| good | 2 | \( 1 + (20.0 - 5.37i)T + (110. - 64i)T^{2} \) |
| 3 | \( 1 + (-16.6 + 62.0i)T + (-1.89e3 - 1.09e3i)T^{2} \) |
| 11 | \( 1 + (-3.09e3 - 5.36e3i)T + (-9.74e6 + 1.68e7i)T^{2} \) |
| 13 | \( 1 + (-1.82e3 - 1.82e3i)T + 6.27e7iT^{2} \) |
| 17 | \( 1 + (-3.11e4 - 8.34e3i)T + (3.55e8 + 2.05e8i)T^{2} \) |
| 19 | \( 1 + (-325. + 564. i)T + (-4.46e8 - 7.74e8i)T^{2} \) |
| 23 | \( 1 + (2.23e4 + 8.34e4i)T + (-2.94e9 + 1.70e9i)T^{2} \) |
| 29 | \( 1 + 3.40e4iT - 1.72e10T^{2} \) |
| 31 | \( 1 + (-1.75e4 + 1.01e4i)T + (1.37e10 - 2.38e10i)T^{2} \) |
| 37 | \( 1 + (1.24e5 - 3.34e4i)T + (8.22e10 - 4.74e10i)T^{2} \) |
| 41 | \( 1 - 2.00e5iT - 1.94e11T^{2} \) |
| 43 | \( 1 + (-3.82e5 + 3.82e5i)T - 2.71e11iT^{2} \) |
| 47 | \( 1 + (4.89e4 + 1.82e5i)T + (-4.38e11 + 2.53e11i)T^{2} \) |
| 53 | \( 1 + (7.62e5 + 2.04e5i)T + (1.01e12 + 5.87e11i)T^{2} \) |
| 59 | \( 1 + (-7.06e5 - 1.22e6i)T + (-1.24e12 + 2.15e12i)T^{2} \) |
| 61 | \( 1 + (2.66e6 + 1.53e6i)T + (1.57e12 + 2.72e12i)T^{2} \) |
| 67 | \( 1 + (-3.38e5 + 1.26e6i)T + (-5.24e12 - 3.03e12i)T^{2} \) |
| 71 | \( 1 - 3.53e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + (-1.50e4 + 5.62e4i)T + (-9.56e12 - 5.52e12i)T^{2} \) |
| 79 | \( 1 + (-2.09e6 - 1.21e6i)T + (9.60e12 + 1.66e13i)T^{2} \) |
| 83 | \( 1 + (-2.03e5 - 2.03e5i)T + 2.71e13iT^{2} \) |
| 89 | \( 1 + (3.06e6 - 5.31e6i)T + (-2.21e13 - 3.83e13i)T^{2} \) |
| 97 | \( 1 + (2.79e6 - 2.79e6i)T - 8.07e13iT^{2} \) |
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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
−14.81074520181909864231462031562, −14.01023817282100132268691813978, −12.26717190211601441665294460273, −10.63312459243977963168300227680, −9.671754062598918228405974788635, −8.125136804698434577240791379615, −7.19547930957182289160754015565, −6.40358592663179041854568017642, −2.02366832286163866031207118393, −1.11841643936025411873524413738,
1.30041516260751598195288697406, 3.16529189440258166598925739515, 5.66057943965653131069489232866, 8.166265966037739535577382869277, 9.132977111236660069651075038461, 9.695510815661184367344270916461, 10.91714770453514126296087950070, 12.06906119478797641494408124992, 14.23650221521036538617662312450, 15.75755155321265848906242193968
