| L(s) = 1 | + (0.849 + 1.01i)2-s + (0.173 + 0.984i)3-s + (0.0442 − 0.250i)4-s + (0.224 − 0.0395i)5-s + (−0.849 + 1.01i)6-s + (1.93 − 1.80i)7-s + (2.57 − 1.48i)8-s + (−0.939 + 0.342i)9-s + (0.230 + 0.193i)10-s + 3.09·11-s + 0.254·12-s + (−0.305 − 0.256i)13-s + (3.46 + 0.423i)14-s + (0.0778 + 0.213i)15-s + (3.21 + 1.17i)16-s + (−1.37 + 3.78i)17-s + ⋯ |
| L(s) = 1 | + (0.600 + 0.715i)2-s + (0.100 + 0.568i)3-s + (0.0221 − 0.125i)4-s + (0.100 − 0.0176i)5-s + (−0.346 + 0.413i)6-s + (0.730 − 0.682i)7-s + (0.912 − 0.526i)8-s + (−0.313 + 0.114i)9-s + (0.0728 + 0.0610i)10-s + 0.933·11-s + 0.0735·12-s + (−0.0847 − 0.0710i)13-s + (0.927 + 0.113i)14-s + (0.0200 + 0.0551i)15-s + (0.804 + 0.292i)16-s + (−0.334 + 0.918i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 399 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.680 - 0.732i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 399 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.680 - 0.732i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.00509 + 0.873847i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.00509 + 0.873847i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.93 + 1.80i)T \) |
| 19 | \( 1 + (3.93 - 1.87i)T \) |
| good | 2 | \( 1 + (-0.849 - 1.01i)T + (-0.347 + 1.96i)T^{2} \) |
| 5 | \( 1 + (-0.224 + 0.0395i)T + (4.69 - 1.71i)T^{2} \) |
| 11 | \( 1 - 3.09T + 11T^{2} \) |
| 13 | \( 1 + (0.305 + 0.256i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (1.37 - 3.78i)T + (-13.0 - 10.9i)T^{2} \) |
| 23 | \( 1 + (0.376 + 0.316i)T + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (-1.68 - 0.297i)T + (27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (1.36 + 2.36i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (1.07 - 0.620i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-3.24 + 2.72i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (3.59 + 1.30i)T + (32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (-2.38 - 6.56i)T + (-36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + (-3.35 - 0.592i)T + (49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (7.12 + 2.59i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (5.10 - 6.08i)T + (-10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (-4.98 + 5.93i)T + (-11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-0.0420 + 0.115i)T + (-54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (7.93 - 1.39i)T + (68.5 - 24.9i)T^{2} \) |
| 79 | \( 1 + (3.80 - 10.4i)T + (-60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (-2.84 - 1.64i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-1.03 + 5.88i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (0.996 + 5.65i)T + (-91.1 + 33.1i)T^{2} \) |
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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
−11.17675352615846446827319112278, −10.54356784544197542533146422703, −9.676577017564391924695094947798, −8.512641692908235488492784708089, −7.54590902635842176545512255983, −6.48037643924686980258672021373, −5.64685429595406572318338286766, −4.42988880710686844168027942956, −3.91734908856430386853402494272, −1.67155174560032302361168525562,
1.75048289029630723560077735265, 2.70844348316775484287074544188, 4.09990523781777947416593240545, 5.09635604237205415712154330436, 6.37908722918403966797808500229, 7.45507484269902329612112280808, 8.418125688801630213792152664303, 9.231907341073471579756370240573, 10.61379389831227074931467117280, 11.62427811511631929250258277066
