| L(s) = 1 | + (1.62 + 1.93i)2-s + (0.173 + 0.984i)3-s + (−0.759 + 4.30i)4-s + (−1.95 + 0.344i)5-s + (−1.62 + 1.93i)6-s + (2.62 + 0.330i)7-s + (−5.18 + 2.99i)8-s + (−0.939 + 0.342i)9-s + (−3.84 − 3.22i)10-s + 0.413·11-s − 4.37·12-s + (−2.21 − 1.85i)13-s + (3.62 + 5.61i)14-s + (−0.679 − 1.86i)15-s + (−5.99 − 2.18i)16-s + (0.435 − 1.19i)17-s + ⋯ |
| L(s) = 1 | + (1.14 + 1.36i)2-s + (0.100 + 0.568i)3-s + (−0.379 + 2.15i)4-s + (−0.874 + 0.154i)5-s + (−0.662 + 0.789i)6-s + (0.992 + 0.124i)7-s + (−1.83 + 1.05i)8-s + (−0.313 + 0.114i)9-s + (−1.21 − 1.01i)10-s + 0.124·11-s − 1.26·12-s + (−0.613 − 0.514i)13-s + (0.967 + 1.49i)14-s + (−0.175 − 0.481i)15-s + (−1.49 − 0.545i)16-s + (0.105 − 0.289i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 399 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.978 - 0.205i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 399 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.978 - 0.205i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.233838 + 2.24712i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.233838 + 2.24712i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.173 - 0.984i)T \) |
| 7 | \( 1 + (-2.62 - 0.330i)T \) |
| 19 | \( 1 + (-4.17 + 1.25i)T \) |
| good | 2 | \( 1 + (-1.62 - 1.93i)T + (-0.347 + 1.96i)T^{2} \) |
| 5 | \( 1 + (1.95 - 0.344i)T + (4.69 - 1.71i)T^{2} \) |
| 11 | \( 1 - 0.413T + 11T^{2} \) |
| 13 | \( 1 + (2.21 + 1.85i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-0.435 + 1.19i)T + (-13.0 - 10.9i)T^{2} \) |
| 23 | \( 1 + (-4.69 - 3.94i)T + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (2.49 + 0.440i)T + (27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (1.18 + 2.05i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-4.33 + 2.50i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (5.86 - 4.91i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-9.48 - 3.45i)T + (32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (2.43 + 6.69i)T + (-36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + (4.95 + 0.874i)T + (49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (-8.94 - 3.25i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-9.02 + 10.7i)T + (-10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (-2.78 + 3.31i)T + (-11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (4.75 - 13.0i)T + (-54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (8.81 - 1.55i)T + (68.5 - 24.9i)T^{2} \) |
| 79 | \( 1 + (-4.15 + 11.4i)T + (-60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (12.3 + 7.14i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-0.587 + 3.33i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (0.230 + 1.30i)T + (-91.1 + 33.1i)T^{2} \) |
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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
−11.67746837982209551673904584915, −11.27689337989886563484827393198, −9.682362449164306787705575797850, −8.504892003702399835189145594842, −7.65744723146491974635377014017, −7.19283672114685664584632811085, −5.64566108248060669976175105884, −5.00394389119752891921685198008, −4.09558544972883932965674105990, −3.07853947977102294540751674084,
1.18102324950068681793605968537, 2.47026070743060438052848674391, 3.76635610751951446792094620602, 4.64209809715263011737345409276, 5.56364547809082240564707590176, 7.05985440598412088107605852444, 8.048147928401500967749485569458, 9.213881452889969302797608303926, 10.41836135937430367602695440653, 11.30044255199921376162932719915
