L(s) = 1 | + (−1.29 − 1.82i)5-s + (0.306 + 0.306i)7-s + 4.06·11-s + (0.625 + 0.625i)13-s + (−3.57 + 3.57i)17-s + 6.82·19-s + (−1.58 − 1.58i)23-s + (−1.65 + 4.71i)25-s − 8.50i·29-s + 2.56·31-s + (0.163 − 0.955i)35-s + (−1.69 + 1.69i)37-s − 5.19i·41-s + (4.18 + 4.18i)43-s + (4.58 − 4.58i)47-s + ⋯ |
L(s) = 1 | + (−0.578 − 0.815i)5-s + (0.115 + 0.115i)7-s + 1.22·11-s + (0.173 + 0.173i)13-s + (−0.867 + 0.867i)17-s + 1.56·19-s + (−0.331 − 0.331i)23-s + (−0.331 + 0.943i)25-s − 1.58i·29-s + 0.461·31-s + (0.0275 − 0.161i)35-s + (−0.277 + 0.277i)37-s − 0.811i·41-s + (0.638 + 0.638i)43-s + (0.668 − 0.668i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.730 + 0.683i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.730 + 0.683i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.609481509\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.609481509\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (1.29 + 1.82i)T \) |
good | 7 | \( 1 + (-0.306 - 0.306i)T + 7iT^{2} \) |
| 11 | \( 1 - 4.06T + 11T^{2} \) |
| 13 | \( 1 + (-0.625 - 0.625i)T + 13iT^{2} \) |
| 17 | \( 1 + (3.57 - 3.57i)T - 17iT^{2} \) |
| 19 | \( 1 - 6.82T + 19T^{2} \) |
| 23 | \( 1 + (1.58 + 1.58i)T + 23iT^{2} \) |
| 29 | \( 1 + 8.50iT - 29T^{2} \) |
| 31 | \( 1 - 2.56T + 31T^{2} \) |
| 37 | \( 1 + (1.69 - 1.69i)T - 37iT^{2} \) |
| 41 | \( 1 + 5.19iT - 41T^{2} \) |
| 43 | \( 1 + (-4.18 - 4.18i)T + 43iT^{2} \) |
| 47 | \( 1 + (-4.58 + 4.58i)T - 47iT^{2} \) |
| 53 | \( 1 + (-7.41 + 7.41i)T - 53iT^{2} \) |
| 59 | \( 1 + 8.79iT - 59T^{2} \) |
| 61 | \( 1 - 6.08iT - 61T^{2} \) |
| 67 | \( 1 + (-6.18 + 6.18i)T - 67iT^{2} \) |
| 71 | \( 1 - 14.7iT - 71T^{2} \) |
| 73 | \( 1 + (-3.05 + 3.05i)T - 73iT^{2} \) |
| 79 | \( 1 + 8.56iT - 79T^{2} \) |
| 83 | \( 1 + (-5.13 + 5.13i)T - 83iT^{2} \) |
| 89 | \( 1 - 9.88T + 89T^{2} \) |
| 97 | \( 1 + (-1.75 - 1.75i)T + 97iT^{2} \) |
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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
−9.312122450546698225474199414892, −8.636865706342606351943817778491, −7.983411739430125485044505463289, −7.01533027450312748625700470016, −6.15821870125806000554132424389, −5.20866311852106782055089517125, −4.20514876212185676326722696546, −3.65703304298600315358649803336, −2.05203608999085736120826211856, −0.820899584193805865063001494083,
1.12225542685509496668279112279, 2.68479167653322675587171753734, 3.56886286083085490772523591885, 4.41399771803622649463831143336, 5.52030616238137418405592819188, 6.57669350561471781756105570370, 7.16441994000945511250058929090, 7.84812341506577139077018283639, 8.982307609511416391161627460949, 9.480339778951955398133124080446