| 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
−11.62427811511631929250258277066, −10.61379389831227074931467117280, −9.231907341073471579756370240573, −8.418125688801630213792152664303, −7.45507484269902329612112280808, −6.37908722918403966797808500229, −5.09635604237205415712154330436, −4.09990523781777947416593240545, −2.70844348316775484287074544188, −1.75048289029630723560077735265,
1.67155174560032302361168525562, 3.91734908856430386853402494272, 4.42988880710686844168027942956, 5.64685429595406572318338286766, 6.48037643924686980258672021373, 7.54590902635842176545512255983, 8.512641692908235488492784708089, 9.676577017564391924695094947798, 10.54356784544197542533146422703, 11.17675352615846446827319112278
