L(s) = 1 | + (0.164 + 0.284i)5-s + (0.5 − 0.866i)7-s + (−0.664 + 1.15i)11-s + (−1.53 − 2.66i)13-s − 7.35·17-s − 2.93·19-s + (−3.34 − 5.79i)23-s + (2.44 − 4.23i)25-s + (−3.88 + 6.72i)29-s + (1.63 + 2.83i)31-s + 0.329·35-s + 0.329·37-s + (0.135 + 0.234i)41-s + (5.48 − 9.49i)43-s + (0.571 − 0.989i)47-s + ⋯ |
L(s) = 1 | + (0.0735 + 0.127i)5-s + (0.188 − 0.327i)7-s + (−0.200 + 0.347i)11-s + (−0.426 − 0.739i)13-s − 1.78·17-s − 0.673·19-s + (−0.697 − 1.20i)23-s + (0.489 − 0.847i)25-s + (−0.720 + 1.24i)29-s + (0.293 + 0.508i)31-s + 0.0556·35-s + 0.0540·37-s + (0.0211 + 0.0366i)41-s + (0.836 − 1.44i)43-s + (0.0832 − 0.144i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.855 + 0.517i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.855 + 0.517i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4989060104\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4989060104\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-0.5 + 0.866i)T \) |
good | 5 | \( 1 + (-0.164 - 0.284i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (0.664 - 1.15i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (1.53 + 2.66i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + 7.35T + 17T^{2} \) |
| 19 | \( 1 + 2.93T + 19T^{2} \) |
| 23 | \( 1 + (3.34 + 5.79i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (3.88 - 6.72i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-1.63 - 2.83i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 0.329T + 37T^{2} \) |
| 41 | \( 1 + (-0.135 - 0.234i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-5.48 + 9.49i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-0.571 + 0.989i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 6.42T + 53T^{2} \) |
| 59 | \( 1 + (-0.372 - 0.644i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (4.42 - 7.65i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (4.28 + 7.42i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 1.60T + 71T^{2} \) |
| 73 | \( 1 + 13.4T + 73T^{2} \) |
| 79 | \( 1 + (-0.628 + 1.08i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (0.0316 - 0.0548i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 11.3T + 89T^{2} \) |
| 97 | \( 1 + (-5.51 + 9.55i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−8.935679700655729473738853784345, −8.508602048290398098113639469959, −7.43975728928448534272867213462, −6.78323478962213696558648313671, −5.93784839517486520509577548676, −4.78099555882421032471928864881, −4.23170324793711605611624182249, −2.86706237907641222765789787290, −1.95925756458330994563788132920, −0.18007897211395220512436647417,
1.74665088151753421873110051210, 2.65132931422994524601180877690, 4.02704699676859783141300458745, 4.70551567011665038656246688836, 5.79763504865198789679576925281, 6.47416898201925401016644853203, 7.46301908383803487555977707120, 8.214606209953547638762797410367, 9.184939445581428059531783056524, 9.501874948852345248825580143321