| L(s) = 1 | + (−22.3 − 5.98i)2-s + (−10.0 − 37.4i)3-s + (20.3 + 11.7i)4-s + (−924. + 1.04e3i)5-s + 896. i·6-s + (−6.33e3 − 518. i)7-s + (7.99e3 + 7.99e3i)8-s + (1.57e4 − 9.09e3i)9-s + (2.69e4 − 1.78e4i)10-s + (−3.04e4 + 5.27e4i)11-s + (235. − 878. i)12-s + (−1.17e5 + 1.17e5i)13-s + (1.38e5 + 4.95e4i)14-s + (4.84e4 + 2.40e4i)15-s + (−1.36e5 − 2.36e5i)16-s + (3.15e5 − 8.45e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.987 − 0.264i)2-s + (−0.0714 − 0.266i)3-s + (0.0397 + 0.0229i)4-s + (−0.661 + 0.749i)5-s + 0.282i·6-s + (−0.996 − 0.0816i)7-s + (0.689 + 0.689i)8-s + (0.799 − 0.461i)9-s + (0.851 − 0.565i)10-s + (−0.627 + 1.08i)11-s + (0.00327 − 0.0122i)12-s + (−1.13 + 1.13i)13-s + (0.962 + 0.344i)14-s + (0.247 + 0.122i)15-s + (−0.521 − 0.903i)16-s + (0.916 − 0.245i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.332 + 0.942i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.332 + 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.364565 - 0.257901i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.364565 - 0.257901i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (924. - 1.04e3i)T \) |
| 7 | \( 1 + (6.33e3 + 518. i)T \) |
| good | 2 | \( 1 + (22.3 + 5.98i)T + (443. + 256i)T^{2} \) |
| 3 | \( 1 + (10.0 + 37.4i)T + (-1.70e4 + 9.84e3i)T^{2} \) |
| 11 | \( 1 + (3.04e4 - 5.27e4i)T + (-1.17e9 - 2.04e9i)T^{2} \) |
| 13 | \( 1 + (1.17e5 - 1.17e5i)T - 1.06e10iT^{2} \) |
| 17 | \( 1 + (-3.15e5 + 8.45e4i)T + (1.02e11 - 5.92e10i)T^{2} \) |
| 19 | \( 1 + (1.44e5 + 2.50e5i)T + (-1.61e11 + 2.79e11i)T^{2} \) |
| 23 | \( 1 + (-1.43e5 + 5.36e5i)T + (-1.55e12 - 9.00e11i)T^{2} \) |
| 29 | \( 1 + 5.36e6iT - 1.45e13T^{2} \) |
| 31 | \( 1 + (8.85e5 + 5.11e5i)T + (1.32e13 + 2.28e13i)T^{2} \) |
| 37 | \( 1 + (-1.12e7 - 3.01e6i)T + (1.12e14 + 6.49e13i)T^{2} \) |
| 41 | \( 1 + 2.35e6iT - 3.27e14T^{2} \) |
| 43 | \( 1 + (9.45e6 + 9.45e6i)T + 5.02e14iT^{2} \) |
| 47 | \( 1 + (1.13e7 - 4.23e7i)T + (-9.69e14 - 5.59e14i)T^{2} \) |
| 53 | \( 1 + (-5.23e6 + 1.40e6i)T + (2.85e15 - 1.64e15i)T^{2} \) |
| 59 | \( 1 + (-4.55e7 + 7.89e7i)T + (-4.33e15 - 7.50e15i)T^{2} \) |
| 61 | \( 1 + (-1.10e8 + 6.35e7i)T + (5.84e15 - 1.01e16i)T^{2} \) |
| 67 | \( 1 + (-5.87e7 - 2.19e8i)T + (-2.35e16 + 1.36e16i)T^{2} \) |
| 71 | \( 1 + 3.71e7T + 4.58e16T^{2} \) |
| 73 | \( 1 + (-2.94e7 - 1.10e8i)T + (-5.09e16 + 2.94e16i)T^{2} \) |
| 79 | \( 1 + (-4.88e8 + 2.81e8i)T + (5.99e16 - 1.03e17i)T^{2} \) |
| 83 | \( 1 + (1.58e8 - 1.58e8i)T - 1.86e17iT^{2} \) |
| 89 | \( 1 + (2.79e8 + 4.84e8i)T + (-1.75e17 + 3.03e17i)T^{2} \) |
| 97 | \( 1 + (9.93e8 + 9.93e8i)T + 7.60e17iT^{2} \) |
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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
−14.41472261222444179544515541339, −12.83417644994299313348605887165, −11.68301408307427502346913691314, −10.07651337233002037762429415933, −9.611433166407246689539846244637, −7.67344301628431822113693325822, −6.83124666550775087731819219597, −4.38967316318893323258489349678, −2.34566163382706335305423406490, −0.37376083403851234900076164213,
0.77639885310964523913146064085, 3.52981170345553432496110529131, 5.24957184100679231134680905780, 7.40161080647003523818382610406, 8.306870704891858104404816687778, 9.654358891404213926912269466408, 10.53595689971844415099168644859, 12.53161583201534580395353206740, 13.19810286739639736998129329987, 15.28155842020940803960910428775
