| L(s) = 1 | + (0.680 − 1.23i)2-s + (0.876 + 0.656i)3-s + (−1.07 − 1.68i)4-s + (−1.79 − 1.32i)5-s + (1.41 − 0.639i)6-s + (3.86 − 0.276i)7-s + (−2.82 + 0.180i)8-s + (−0.507 − 1.72i)9-s + (−2.87 + 1.32i)10-s + (−0.203 − 0.316i)11-s + (0.167 − 2.18i)12-s + (0.0210 − 0.294i)13-s + (2.28 − 4.97i)14-s + (−0.706 − 2.34i)15-s + (−1.69 + 3.62i)16-s + (−2.18 − 5.84i)17-s + ⋯ |
| L(s) = 1 | + (0.481 − 0.876i)2-s + (0.506 + 0.378i)3-s + (−0.536 − 0.843i)4-s + (−0.804 − 0.593i)5-s + (0.575 − 0.261i)6-s + (1.46 − 0.104i)7-s + (−0.997 + 0.0637i)8-s + (−0.169 − 0.576i)9-s + (−0.907 + 0.419i)10-s + (−0.0613 − 0.0954i)11-s + (0.0482 − 0.630i)12-s + (0.00583 − 0.0815i)13-s + (0.611 − 1.33i)14-s + (−0.182 − 0.605i)15-s + (−0.424 + 0.905i)16-s + (−0.528 − 1.41i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.430 + 0.902i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.430 + 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.980983 - 1.55472i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.980983 - 1.55472i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.680 + 1.23i)T \) |
| 5 | \( 1 + (1.79 + 1.32i)T \) |
| 23 | \( 1 + (-4.04 + 2.57i)T \) |
| good | 3 | \( 1 + (-0.876 - 0.656i)T + (0.845 + 2.87i)T^{2} \) |
| 7 | \( 1 + (-3.86 + 0.276i)T + (6.92 - 0.996i)T^{2} \) |
| 11 | \( 1 + (0.203 + 0.316i)T + (-4.56 + 10.0i)T^{2} \) |
| 13 | \( 1 + (-0.0210 + 0.294i)T + (-12.8 - 1.85i)T^{2} \) |
| 17 | \( 1 + (2.18 + 5.84i)T + (-12.8 + 11.1i)T^{2} \) |
| 19 | \( 1 + (-1.39 - 3.05i)T + (-12.4 + 14.3i)T^{2} \) |
| 29 | \( 1 + (3.83 + 1.75i)T + (18.9 + 21.9i)T^{2} \) |
| 31 | \( 1 + (2.71 + 0.389i)T + (29.7 + 8.73i)T^{2} \) |
| 37 | \( 1 + (3.43 - 6.29i)T + (-20.0 - 31.1i)T^{2} \) |
| 41 | \( 1 + (-10.4 - 3.08i)T + (34.4 + 22.1i)T^{2} \) |
| 43 | \( 1 + (-1.44 - 1.08i)T + (12.1 + 41.2i)T^{2} \) |
| 47 | \( 1 + (-0.303 - 0.303i)T + 47iT^{2} \) |
| 53 | \( 1 + (-7.03 + 0.503i)T + (52.4 - 7.54i)T^{2} \) |
| 59 | \( 1 + (-7.04 - 8.13i)T + (-8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (-0.209 + 1.45i)T + (-58.5 - 17.1i)T^{2} \) |
| 67 | \( 1 + (-2.96 - 13.6i)T + (-60.9 + 27.8i)T^{2} \) |
| 71 | \( 1 + (-1.89 + 2.94i)T + (-29.4 - 64.5i)T^{2} \) |
| 73 | \( 1 + (3.18 - 8.53i)T + (-55.1 - 47.8i)T^{2} \) |
| 79 | \( 1 + (3.89 + 4.49i)T + (-11.2 + 78.1i)T^{2} \) |
| 83 | \( 1 + (3.11 - 5.69i)T + (-44.8 - 69.8i)T^{2} \) |
| 89 | \( 1 + (8.15 - 1.17i)T + (85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (0.999 - 0.545i)T + (52.4 - 81.6i)T^{2} \) |
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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
−11.17088674877068515327800778097, −9.902415805087912352840480124598, −8.968656699657971460939064135442, −8.400234434987189324974843660763, −7.25737458966113122132029091116, −5.54117882016022882534078456295, −4.65416311117959045813227085542, −3.93777922542647144933389175587, −2.69431530375853809693899985572, −1.01772771794835696193527337109,
2.20679500766247069763429348220, 3.63054937576598500666342672739, 4.67076867710107628897753301733, 5.63498250283049402316031881368, 7.08739290328530801790422166389, 7.57280432047690092966827967009, 8.349289557079224573618590298881, 8.990551742256332382250937838699, 10.88276890176334452460180385361, 11.22835913317295357555858367609
