L(s) = 1 | + (−0.536 + 2.00i)3-s + (−0.829 + 2.07i)5-s + (−2.33 + 1.24i)7-s + (−1.11 − 0.644i)9-s + (−3.09 − 5.36i)11-s + (−0.782 − 0.782i)13-s + (−3.70 − 2.77i)15-s + (4.40 + 1.18i)17-s + (−2.37 + 4.11i)19-s + (−1.25 − 5.33i)21-s + (1.74 + 6.52i)23-s + (−3.62 − 3.44i)25-s + (−2.50 + 2.50i)27-s + 5.30i·29-s + (2.16 − 1.25i)31-s + ⋯ |
L(s) = 1 | + (−0.309 + 1.15i)3-s + (−0.370 + 0.928i)5-s + (−0.881 + 0.472i)7-s + (−0.372 − 0.214i)9-s + (−0.933 − 1.61i)11-s + (−0.217 − 0.217i)13-s + (−0.957 − 0.715i)15-s + (1.06 + 0.286i)17-s + (−0.544 + 0.943i)19-s + (−0.272 − 1.16i)21-s + (0.364 + 1.35i)23-s + (−0.724 − 0.688i)25-s + (−0.482 + 0.482i)27-s + 0.985i·29-s + (0.389 − 0.224i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.969 - 0.243i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.969 - 0.243i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0858820 + 0.693391i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0858820 + 0.693391i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.829 - 2.07i)T \) |
| 7 | \( 1 + (2.33 - 1.24i)T \) |
good | 3 | \( 1 + (0.536 - 2.00i)T + (-2.59 - 1.5i)T^{2} \) |
| 11 | \( 1 + (3.09 + 5.36i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (0.782 + 0.782i)T + 13iT^{2} \) |
| 17 | \( 1 + (-4.40 - 1.18i)T + (14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (2.37 - 4.11i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.74 - 6.52i)T + (-19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 - 5.30iT - 29T^{2} \) |
| 31 | \( 1 + (-2.16 + 1.25i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (6.61 - 1.77i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 - 2.51iT - 41T^{2} \) |
| 43 | \( 1 + (-0.404 + 0.404i)T - 43iT^{2} \) |
| 47 | \( 1 + (-2.31 - 8.63i)T + (-40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (-9.92 - 2.66i)T + (45.8 + 26.5i)T^{2} \) |
| 59 | \( 1 + (0.0710 + 0.123i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.67 - 1.54i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.16 + 4.34i)T + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 - 10.3T + 71T^{2} \) |
| 73 | \( 1 + (0.483 - 1.80i)T + (-63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (5.42 + 3.13i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (5.82 + 5.82i)T + 83iT^{2} \) |
| 89 | \( 1 + (-4.58 + 7.94i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (2.64 - 2.64i)T - 97iT^{2} \) |
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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
−12.09125918213367932487653921435, −11.06159961480159904302572052814, −10.42755036635804574116448954525, −9.788392744115492077761311227632, −8.555220437322587150264020448588, −7.47958038414813851187691031574, −6.03520127200216721824994425294, −5.38999587559554500159179426933, −3.65625137439770895887089109385, −3.08097765921681060290281806757,
0.53528494989144854078230000363, 2.29832228432794155329773610556, 4.22473826890125687648764454750, 5.32951148130624708684518176780, 6.80844384556410357028633372173, 7.28818853148917752364292615738, 8.328529116273957143239266530520, 9.599905681587148729397656298970, 10.38881090928410506674387933540, 11.89953450339838216925898081990