| L(s) = 1 | + (−0.366 − 1.36i)2-s + (−1.76 − 2.42i)3-s + (−1.73 + i)4-s + (2.77 + 4.15i)5-s + (−2.67 + 3.29i)6-s + (−5.04 − 4.85i)7-s + (2 + 1.99i)8-s + (−2.80 + 8.55i)9-s + (4.66 − 5.31i)10-s + (−17.0 + 9.81i)11-s + (5.47 + 2.44i)12-s + (7.73 + 7.73i)13-s + (−4.78 + 8.66i)14-s + (5.21 − 14.0i)15-s + (1.99 − 3.46i)16-s + (−3.28 + 12.2i)17-s + ⋯ |
| L(s) = 1 | + (−0.183 − 0.683i)2-s + (−0.586 − 0.809i)3-s + (−0.433 + 0.250i)4-s + (0.554 + 0.831i)5-s + (−0.445 + 0.549i)6-s + (−0.720 − 0.693i)7-s + (0.250 + 0.249i)8-s + (−0.311 + 0.950i)9-s + (0.466 − 0.531i)10-s + (−1.54 + 0.892i)11-s + (0.456 + 0.203i)12-s + (0.594 + 0.594i)13-s + (−0.342 + 0.618i)14-s + (0.347 − 0.937i)15-s + (0.124 − 0.216i)16-s + (−0.193 + 0.721i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.420 - 0.907i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.420 - 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.418254 + 0.267265i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.418254 + 0.267265i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.366 + 1.36i)T \) |
| 3 | \( 1 + (1.76 + 2.42i)T \) |
| 5 | \( 1 + (-2.77 - 4.15i)T \) |
| 7 | \( 1 + (5.04 + 4.85i)T \) |
| good | 11 | \( 1 + (17.0 - 9.81i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (-7.73 - 7.73i)T + 169iT^{2} \) |
| 17 | \( 1 + (3.28 - 12.2i)T + (-250. - 144.5i)T^{2} \) |
| 19 | \( 1 + (0.306 - 0.530i)T + (-180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (-12.1 + 3.26i)T + (458. - 264.5i)T^{2} \) |
| 29 | \( 1 - 29.8T + 841T^{2} \) |
| 31 | \( 1 + (17.4 - 10.0i)T + (480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + (-13.2 + 3.54i)T + (1.18e3 - 684.5i)T^{2} \) |
| 41 | \( 1 + 75.0T + 1.68e3T^{2} \) |
| 43 | \( 1 + (46.7 - 46.7i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (50.2 - 13.4i)T + (1.91e3 - 1.10e3i)T^{2} \) |
| 53 | \( 1 + (-17.2 + 64.3i)T + (-2.43e3 - 1.40e3i)T^{2} \) |
| 59 | \( 1 + (64.0 - 36.9i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (19.2 + 11.1i)T + (1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (2.36 - 8.82i)T + (-3.88e3 - 2.24e3i)T^{2} \) |
| 71 | \( 1 + 4.20iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (-6.15 + 22.9i)T + (-4.61e3 - 2.66e3i)T^{2} \) |
| 79 | \( 1 + (-67.0 - 38.7i)T + (3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-41.1 + 41.1i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 + (-1.42 - 0.825i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (92.5 - 92.5i)T - 9.40e3iT^{2} \) |
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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
−12.40387366151366169868535488972, −11.15306331536935490468004169054, −10.49475818889638764288676871310, −9.834557174270807599001309923114, −8.222260455177998704243375249772, −7.10918449321761907510450862137, −6.33534996136887839513266000257, −4.89678703571234451650315420068, −3.09855305532230093229834569734, −1.78752367803481900195070457187,
0.30709765197987339558387897089, 3.14336053995718690216893200865, 5.01088325270605549951520397117, 5.52864573945722164300329033199, 6.46887365293341245927641402106, 8.243665447848953075360101395769, 8.969274958461679410081218867708, 9.915871025169638527508368717028, 10.71301471411126932145236281338, 12.01095349430286345820628518714
