L(s) = 1 | + (−0.767 + 1.32i)5-s − 2.31i·7-s − 5.49i·11-s + (−3.96 + 2.29i)13-s + (−2.79 + 4.84i)17-s + (1.09 + 4.21i)19-s + (−3.07 + 1.77i)23-s + (1.32 + 2.29i)25-s + (7.12 − 4.11i)29-s + 6.20·31-s + (3.07 + 1.77i)35-s + 1.73i·37-s + (−5.86 − 3.38i)41-s + (9.30 + 5.37i)43-s + (1.68 − 0.973i)47-s + ⋯ |
L(s) = 1 | + (−0.343 + 0.594i)5-s − 0.874i·7-s − 1.65i·11-s + (−1.10 + 0.635i)13-s + (−0.678 + 1.17i)17-s + (0.252 + 0.967i)19-s + (−0.641 + 0.370i)23-s + (0.264 + 0.458i)25-s + (1.32 − 0.764i)29-s + 1.11·31-s + (0.519 + 0.300i)35-s + 0.284i·37-s + (−0.916 − 0.528i)41-s + (1.41 + 0.819i)43-s + (0.246 − 0.142i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.938 - 0.345i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.938 - 0.345i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.431453008\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.431453008\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (-1.09 - 4.21i)T \) |
good | 5 | \( 1 + (0.767 - 1.32i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + 2.31iT - 7T^{2} \) |
| 11 | \( 1 + 5.49iT - 11T^{2} \) |
| 13 | \( 1 + (3.96 - 2.29i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (2.79 - 4.84i)T + (-8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (3.07 - 1.77i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-7.12 + 4.11i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 6.20T + 31T^{2} \) |
| 37 | \( 1 - 1.73iT - 37T^{2} \) |
| 41 | \( 1 + (5.86 + 3.38i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-9.30 - 5.37i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-1.68 + 0.973i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-8.16 + 4.71i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (4.76 - 8.24i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (2.32 + 4.02i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-4.20 - 7.27i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-7.83 + 13.5i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-6.79 + 11.7i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-7.11 + 12.3i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 1.94iT - 83T^{2} \) |
| 89 | \( 1 + (-4.82 + 2.78i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-7.40 - 4.27i)T + (48.5 + 84.0i)T^{2} \) |
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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
−8.743664969330711746565640057208, −8.017124967201227051171514698166, −7.45807871854682921257991217559, −6.46486307374649739681837617767, −6.07301041131961025375295949490, −4.84641169308433897037292236552, −3.94981426547256047699275248966, −3.35147625003728344070629011053, −2.23578917801768846090931087539, −0.824206617616686873051228439753,
0.65035740500995413944162787013, 2.33418766214750087632839959130, 2.70792942523064930389814426257, 4.37968627529967897965919481063, 4.79797576214960046875381590124, 5.43265131113214075298245073091, 6.74917313402787002461423736880, 7.15900807930752126661753512212, 8.118378359077535286773541336544, 8.769751546728099074570921754903