| L(s) = 1 | + (1.04 + 0.955i)2-s + (−0.453 + 0.891i)3-s + (0.174 + 1.99i)4-s + (0.328 + 2.21i)5-s + (−1.32 + 0.495i)6-s + (3.17 − 3.17i)7-s + (−1.72 + 2.24i)8-s + (−0.587 − 0.809i)9-s + (−1.77 + 2.61i)10-s + (−1.22 + 1.68i)11-s + (−1.85 − 0.749i)12-s + (−0.339 + 2.14i)13-s + (6.34 − 0.277i)14-s + (−2.11 − 0.711i)15-s + (−3.93 + 0.694i)16-s + (−2.26 + 1.15i)17-s + ⋯ |
| L(s) = 1 | + (0.737 + 0.675i)2-s + (−0.262 + 0.514i)3-s + (0.0871 + 0.996i)4-s + (0.146 + 0.989i)5-s + (−0.540 + 0.202i)6-s + (1.20 − 1.20i)7-s + (−0.608 + 0.793i)8-s + (−0.195 − 0.269i)9-s + (−0.560 + 0.828i)10-s + (−0.368 + 0.507i)11-s + (−0.535 − 0.216i)12-s + (−0.0942 + 0.594i)13-s + (1.69 − 0.0740i)14-s + (−0.547 − 0.183i)15-s + (−0.984 + 0.173i)16-s + (−0.548 + 0.279i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.401 - 0.915i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.401 - 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.01490 + 1.55314i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.01490 + 1.55314i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.04 - 0.955i)T \) |
| 3 | \( 1 + (0.453 - 0.891i)T \) |
| 5 | \( 1 + (-0.328 - 2.21i)T \) |
| good | 7 | \( 1 + (-3.17 + 3.17i)T - 7iT^{2} \) |
| 11 | \( 1 + (1.22 - 1.68i)T + (-3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (0.339 - 2.14i)T + (-12.3 - 4.01i)T^{2} \) |
| 17 | \( 1 + (2.26 - 1.15i)T + (9.99 - 13.7i)T^{2} \) |
| 19 | \( 1 + (-1.87 + 5.78i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (0.703 + 4.44i)T + (-21.8 + 7.10i)T^{2} \) |
| 29 | \( 1 + (-1.37 + 0.445i)T + (23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-1.68 - 0.546i)T + (25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-8.87 - 1.40i)T + (35.1 + 11.4i)T^{2} \) |
| 41 | \( 1 + (-5.90 + 4.28i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + (-4.19 - 4.19i)T + 43iT^{2} \) |
| 47 | \( 1 + (-1.82 - 0.928i)T + (27.6 + 38.0i)T^{2} \) |
| 53 | \( 1 + (-10.2 - 5.24i)T + (31.1 + 42.8i)T^{2} \) |
| 59 | \( 1 + (3.92 - 2.84i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (11.1 + 8.09i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + (3.16 + 6.20i)T + (-39.3 + 54.2i)T^{2} \) |
| 71 | \( 1 + (8.47 - 2.75i)T + (57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (0.968 - 0.153i)T + (69.4 - 22.5i)T^{2} \) |
| 79 | \( 1 + (4.20 + 12.9i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (4.17 - 2.12i)T + (48.7 - 67.1i)T^{2} \) |
| 89 | \( 1 + (-9.07 + 12.4i)T + (-27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (4.76 - 9.35i)T + (-57.0 - 78.4i)T^{2} \) |
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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
−11.89770117830197007496923725336, −11.06605006444756878786713352470, −10.53394457517135791777037439968, −9.146641544562088391722528618069, −7.78585625452852022931833657013, −7.14020463625802037621352880675, −6.16741325783901443179101763676, −4.70466642342085347978875890787, −4.22907234819706105578800759493, −2.59967627653891464628719793952,
1.33371980568565018263108966349, 2.56112350915124015019799914893, 4.39259067902588226139312415267, 5.58396116278930935660697408779, 5.74539658181843898939797232058, 7.74116050190525270627076409937, 8.611187006372270403997276028990, 9.635836847424535828839322816846, 10.90164086182657284379203776363, 11.76195151212150846090260603644
