| L(s) = 1 | + (−0.766 + 0.642i)2-s + (1.46 + 0.531i)3-s + (0.173 − 0.984i)4-s + (−0.712 − 4.04i)5-s + (−1.46 + 0.531i)6-s + (−0.291 + 0.505i)7-s + (0.500 + 0.866i)8-s + (−0.445 − 0.373i)9-s + (3.14 + 2.63i)10-s + (−0.5 − 0.866i)11-s + (0.777 − 1.34i)12-s + (−1.18 + 0.432i)13-s + (−0.101 − 0.574i)14-s + (1.10 − 6.28i)15-s + (−0.939 − 0.342i)16-s + (2.52 − 2.11i)17-s + ⋯ |
| L(s) = 1 | + (−0.541 + 0.454i)2-s + (0.843 + 0.307i)3-s + (0.0868 − 0.492i)4-s + (−0.318 − 1.80i)5-s + (−0.596 + 0.217i)6-s + (−0.110 + 0.190i)7-s + (0.176 + 0.306i)8-s + (−0.148 − 0.124i)9-s + (0.994 + 0.834i)10-s + (−0.150 − 0.261i)11-s + (0.224 − 0.388i)12-s + (−0.329 + 0.119i)13-s + (−0.0270 − 0.153i)14-s + (0.286 − 1.62i)15-s + (−0.234 − 0.0855i)16-s + (0.611 − 0.513i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.413 + 0.910i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.413 + 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.929608 - 0.598523i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.929608 - 0.598523i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.766 - 0.642i)T \) |
| 11 | \( 1 + (0.5 + 0.866i)T \) |
| 19 | \( 1 + (3.09 + 3.06i)T \) |
| good | 3 | \( 1 + (-1.46 - 0.531i)T + (2.29 + 1.92i)T^{2} \) |
| 5 | \( 1 + (0.712 + 4.04i)T + (-4.69 + 1.71i)T^{2} \) |
| 7 | \( 1 + (0.291 - 0.505i)T + (-3.5 - 6.06i)T^{2} \) |
| 13 | \( 1 + (1.18 - 0.432i)T + (9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (-2.52 + 2.11i)T + (2.95 - 16.7i)T^{2} \) |
| 23 | \( 1 + (-1.35 + 7.66i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (-7.36 - 6.17i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (-0.00593 + 0.0102i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 5.09T + 37T^{2} \) |
| 41 | \( 1 + (-2.24 - 0.818i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (-0.0356 - 0.202i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (-2.30 - 1.93i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + (2.07 - 11.7i)T + (-49.8 - 18.1i)T^{2} \) |
| 59 | \( 1 + (-7.70 + 6.46i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (-1.31 + 7.45i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (10.0 + 8.42i)T + (11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (-0.249 - 1.41i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (-0.954 - 0.347i)T + (55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (-13.4 - 4.89i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (5.71 - 9.90i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-0.710 + 0.258i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (3.59 - 3.01i)T + (16.8 - 95.5i)T^{2} \) |
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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
−10.87127254885127317715093538381, −9.637888382742402464076965725059, −8.995755005769678862433009073261, −8.512929011261218949066266325469, −7.78132767962806837991890217518, −6.35679856254694183773102926303, −5.10687193967452269117757242764, −4.33023879666852925978370134712, −2.68682130546055607830413364726, −0.77150516941854374390620718743,
2.13412372206578062339746721844, 3.01434726190775960328036493884, 3.88090486759444690777393256121, 5.94641176683969015865432263554, 7.13195906586976398104269047140, 7.69753973276426847973739979815, 8.467998122970218143566760075217, 9.871304973584053604193445098241, 10.29872692210819342630518504506, 11.25921227159451609574759984010
