| L(s) = 1 | + (0.768 + 1.50i)2-s + (−0.510 + 0.702i)4-s + (−0.466 − 2.94i)5-s + (−0.00626 + 0.00534i)7-s + (1.89 + 0.299i)8-s + (4.08 − 2.96i)10-s + (0.330 + 0.202i)11-s + (5.80 − 0.456i)13-s + (−0.0128 − 0.00533i)14-s + (1.53 + 4.73i)16-s + (−4.08 + 0.981i)17-s + (5.19 + 0.408i)19-s + (2.30 + 1.17i)20-s + (−0.0515 + 0.655i)22-s + (0.926 − 2.85i)23-s + ⋯ |
| L(s) = 1 | + (0.543 + 1.06i)2-s + (−0.255 + 0.351i)4-s + (−0.208 − 1.31i)5-s + (−0.00236 + 0.00202i)7-s + (0.669 + 0.106i)8-s + (1.29 − 0.938i)10-s + (0.0997 + 0.0611i)11-s + (1.60 − 0.126i)13-s + (−0.00344 − 0.00142i)14-s + (0.384 + 1.18i)16-s + (−0.991 + 0.238i)17-s + (1.19 + 0.0937i)19-s + (0.515 + 0.262i)20-s + (−0.0109 + 0.139i)22-s + (0.193 − 0.594i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 369 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.867 - 0.497i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 369 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.867 - 0.497i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.85480 + 0.494066i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.85480 + 0.494066i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 41 | \( 1 + (4.17 - 4.85i)T \) |
| good | 2 | \( 1 + (-0.768 - 1.50i)T + (-1.17 + 1.61i)T^{2} \) |
| 5 | \( 1 + (0.466 + 2.94i)T + (-4.75 + 1.54i)T^{2} \) |
| 7 | \( 1 + (0.00626 - 0.00534i)T + (1.09 - 6.91i)T^{2} \) |
| 11 | \( 1 + (-0.330 - 0.202i)T + (4.99 + 9.80i)T^{2} \) |
| 13 | \( 1 + (-5.80 + 0.456i)T + (12.8 - 2.03i)T^{2} \) |
| 17 | \( 1 + (4.08 - 0.981i)T + (15.1 - 7.71i)T^{2} \) |
| 19 | \( 1 + (-5.19 - 0.408i)T + (18.7 + 2.97i)T^{2} \) |
| 23 | \( 1 + (-0.926 + 2.85i)T + (-18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (5.98 + 1.43i)T + (25.8 + 13.1i)T^{2} \) |
| 31 | \( 1 + (1.86 + 2.56i)T + (-9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (3.92 + 2.84i)T + (11.4 + 35.1i)T^{2} \) |
| 43 | \( 1 + (9.72 - 4.95i)T + (25.2 - 34.7i)T^{2} \) |
| 47 | \( 1 + (-4.66 + 5.46i)T + (-7.35 - 46.4i)T^{2} \) |
| 53 | \( 1 + (2.53 - 10.5i)T + (-47.2 - 24.0i)T^{2} \) |
| 59 | \( 1 + (0.310 + 0.100i)T + (47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (2.22 - 4.37i)T + (-35.8 - 49.3i)T^{2} \) |
| 67 | \( 1 + (-0.450 + 0.276i)T + (30.4 - 59.6i)T^{2} \) |
| 71 | \( 1 + (-5.72 + 9.33i)T + (-32.2 - 63.2i)T^{2} \) |
| 73 | \( 1 + (0.536 + 0.536i)T + 73iT^{2} \) |
| 79 | \( 1 + (0.248 - 0.102i)T + (55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 - 13.5iT - 83T^{2} \) |
| 89 | \( 1 + (-3.01 - 3.53i)T + (-13.9 + 87.9i)T^{2} \) |
| 97 | \( 1 + (4.26 + 6.96i)T + (-44.0 + 86.4i)T^{2} \) |
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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
−11.54358439114256770821251115815, −10.68842710080983444783711386393, −9.275403696326417273142510101321, −8.510412480935402019721471968382, −7.70683464093798174741826163331, −6.52355525092296129968789118996, −5.65507361375511788370455590840, −4.76379105737588485924553493444, −3.80928528884175626490340824774, −1.41042724414088578923978597667,
1.80020192294547663952477230969, 3.25648578403822143107104783485, 3.69252216177666143617402974456, 5.24147763384185055275273091710, 6.64773328428406372954759015252, 7.34849101047492620443954177137, 8.670117719566055821717515540320, 9.895016295852281012671481583077, 10.84914874397949676605420129544, 11.25673930039083517306881233143
