L(s) = 1 | + (−0.364 − 1.35i)3-s + (−0.489 + 2.18i)5-s + (−0.501 − 2.59i)7-s + (0.882 − 0.509i)9-s + (1.86 − 3.23i)11-s + (4.55 − 4.55i)13-s + (3.14 − 0.129i)15-s + (−5.91 + 1.58i)17-s + (0.616 + 1.06i)19-s + (−3.34 + 1.62i)21-s + (0.721 − 2.69i)23-s + (−4.52 − 2.13i)25-s + (−3.99 − 3.99i)27-s + 2.82i·29-s + (5.94 + 3.43i)31-s + ⋯ |
L(s) = 1 | + (−0.210 − 0.784i)3-s + (−0.218 + 0.975i)5-s + (−0.189 − 0.981i)7-s + (0.294 − 0.169i)9-s + (0.563 − 0.975i)11-s + (1.26 − 1.26i)13-s + (0.811 − 0.0333i)15-s + (−1.43 + 0.384i)17-s + (0.141 + 0.244i)19-s + (−0.730 + 0.355i)21-s + (0.150 − 0.561i)23-s + (−0.904 − 0.427i)25-s + (−0.769 − 0.769i)27-s + 0.525i·29-s + (1.06 + 0.616i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.322 + 0.946i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.322 + 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.951415 - 0.680899i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.951415 - 0.680899i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.489 - 2.18i)T \) |
| 7 | \( 1 + (0.501 + 2.59i)T \) |
good | 3 | \( 1 + (0.364 + 1.35i)T + (-2.59 + 1.5i)T^{2} \) |
| 11 | \( 1 + (-1.86 + 3.23i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-4.55 + 4.55i)T - 13iT^{2} \) |
| 17 | \( 1 + (5.91 - 1.58i)T + (14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (-0.616 - 1.06i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.721 + 2.69i)T + (-19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 - 2.82iT - 29T^{2} \) |
| 31 | \( 1 + (-5.94 - 3.43i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-7.52 - 2.01i)T + (32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 - 5.56iT - 41T^{2} \) |
| 43 | \( 1 + (1.95 + 1.95i)T + 43iT^{2} \) |
| 47 | \( 1 + (2.98 - 11.1i)T + (-40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-9.44 + 2.53i)T + (45.8 - 26.5i)T^{2} \) |
| 59 | \( 1 + (-0.916 + 1.58i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (3.43 - 1.98i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (2.12 + 7.93i)T + (-58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 - 0.570T + 71T^{2} \) |
| 73 | \( 1 + (0.546 + 2.03i)T + (-63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (9.47 - 5.47i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (8.80 - 8.80i)T - 83iT^{2} \) |
| 89 | \( 1 + (-3.00 - 5.20i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (3.70 + 3.70i)T + 97iT^{2} \) |
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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
−11.45114443967298720447707225544, −10.88674920780437387053288132949, −10.07388723629347228429507875542, −8.546796026047725482829358033321, −7.67210528027738303967224513860, −6.51637749235625177827238221415, −6.24741308956382659686060551633, −4.14357616339056791717272514644, −3.10163954149736761816763619683, −1.02325293313171568007617336136,
1.92724897853119836866160696668, 4.08937482648178341937602802444, 4.63801585247762892884673765995, 5.89830314879483199636745406885, 7.07855853997514519958815631539, 8.631004751683517967543759503060, 9.172424609608034681614342378597, 9.917072548031691146088620753585, 11.43851170074421311468739826660, 11.72245744206586662965861273385