L(s) = 1 | + (−0.514 + 2.71i)2-s + (0.0531 − 1.42i)3-s + (−5.25 − 2.06i)4-s + (−2.17 − 0.504i)5-s + (3.83 + 0.875i)6-s + (2.44 − 1.01i)7-s + (5.37 − 8.54i)8-s + (0.974 + 0.0730i)9-s + (2.49 − 5.66i)10-s + (−0.0463 − 0.618i)11-s + (−3.21 + 7.36i)12-s + (0.334 − 0.116i)13-s + (1.50 + 7.16i)14-s + (−0.832 + 3.06i)15-s + (12.1 + 11.3i)16-s + (−1.29 − 1.75i)17-s + ⋯ |
L(s) = 1 | + (−0.363 + 1.92i)2-s + (0.0307 − 0.820i)3-s + (−2.62 − 1.03i)4-s + (−0.974 − 0.225i)5-s + (1.56 + 0.357i)6-s + (0.923 − 0.383i)7-s + (1.89 − 3.02i)8-s + (0.324 + 0.0243i)9-s + (0.787 − 1.79i)10-s + (−0.0139 − 0.186i)11-s + (−0.927 + 2.12i)12-s + (0.0927 − 0.0324i)13-s + (0.402 + 1.91i)14-s + (−0.214 + 0.792i)15-s + (3.04 + 2.82i)16-s + (−0.314 − 0.425i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.956 - 0.291i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.956 - 0.291i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.773656 + 0.115239i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.773656 + 0.115239i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (2.17 + 0.504i)T \) |
| 7 | \( 1 + (-2.44 + 1.01i)T \) |
good | 2 | \( 1 + (0.514 - 2.71i)T + (-1.86 - 0.730i)T^{2} \) |
| 3 | \( 1 + (-0.0531 + 1.42i)T + (-2.99 - 0.224i)T^{2} \) |
| 11 | \( 1 + (0.0463 + 0.618i)T + (-10.8 + 1.63i)T^{2} \) |
| 13 | \( 1 + (-0.334 + 0.116i)T + (10.1 - 8.10i)T^{2} \) |
| 17 | \( 1 + (1.29 + 1.75i)T + (-5.01 + 16.2i)T^{2} \) |
| 19 | \( 1 + (-3.50 + 6.07i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (5.29 + 3.90i)T + (6.77 + 21.9i)T^{2} \) |
| 29 | \( 1 + (-0.477 - 0.380i)T + (6.45 + 28.2i)T^{2} \) |
| 31 | \( 1 + (-3.02 + 1.74i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.631 - 0.275i)T + (25.1 + 27.1i)T^{2} \) |
| 41 | \( 1 + (2.54 - 0.579i)T + (36.9 - 17.7i)T^{2} \) |
| 43 | \( 1 + (6.37 - 4.00i)T + (18.6 - 38.7i)T^{2} \) |
| 47 | \( 1 + (-4.39 - 0.832i)T + (43.7 + 17.1i)T^{2} \) |
| 53 | \( 1 + (8.85 - 3.86i)T + (36.0 - 38.8i)T^{2} \) |
| 59 | \( 1 + (-6.11 + 1.88i)T + (48.7 - 33.2i)T^{2} \) |
| 61 | \( 1 + (1.75 - 0.688i)T + (44.7 - 41.4i)T^{2} \) |
| 67 | \( 1 + (3.93 + 1.05i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (-9.82 - 12.3i)T + (-15.7 + 69.2i)T^{2} \) |
| 73 | \( 1 + (-7.05 + 1.33i)T + (67.9 - 26.6i)T^{2} \) |
| 79 | \( 1 + (-6.35 - 3.66i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-0.113 + 0.324i)T + (-64.8 - 51.7i)T^{2} \) |
| 89 | \( 1 + (0.604 - 8.06i)T + (-88.0 - 13.2i)T^{2} \) |
| 97 | \( 1 + (10.6 + 10.6i)T + 97iT^{2} \) |
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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
−12.43536437225461701410682674385, −11.18724677128280514283341419148, −9.803109030696379468276499571487, −8.568632879238651702260826659139, −7.967065094825235736347378758599, −7.26135322012554037746555444033, −6.53426440914981354960902128351, −5.02948498612205728778898367746, −4.26892576469214899264547392453, −0.808678918880526564443468100001,
1.71673350269761319333919453035, 3.43616536524036685665833269067, 4.14777134251910980502400474535, 5.12898268016602076167365197886, 7.76865638036622207886162663294, 8.436996440731501372302104545446, 9.557091918480785241782902996323, 10.32519913547847505277191621999, 11.05012323765542703645695199728, 11.92237563770635793529307357945