| L(s) = 1 | + (0.390 + 0.920i)2-s + (0.783 − 0.152i)3-s + (−0.694 + 0.719i)4-s + (0.394 + 2.20i)5-s + (0.446 + 0.661i)6-s + (0.314 − 1.17i)7-s + (−0.933 − 0.358i)8-s + (−2.19 + 0.885i)9-s + (−1.87 + 1.22i)10-s + (−4.55 − 0.969i)11-s + (−0.434 + 0.669i)12-s + (−1.74 + 0.214i)13-s + (1.20 − 0.169i)14-s + (0.643 + 1.66i)15-s + (−0.0348 − 0.999i)16-s + (−2.38 + 3.96i)17-s + ⋯ |
| L(s) = 1 | + (0.276 + 0.650i)2-s + (0.452 − 0.0878i)3-s + (−0.347 + 0.359i)4-s + (0.176 + 0.984i)5-s + (0.182 + 0.270i)6-s + (0.118 − 0.443i)7-s + (−0.330 − 0.126i)8-s + (−0.730 + 0.295i)9-s + (−0.592 + 0.386i)10-s + (−1.37 − 0.292i)11-s + (−0.125 + 0.193i)12-s + (−0.484 + 0.0595i)13-s + (0.321 − 0.0451i)14-s + (0.166 + 0.429i)15-s + (−0.00872 − 0.249i)16-s + (−0.577 + 0.961i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0420i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0420i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0233778 + 1.11251i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0233778 + 1.11251i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.390 - 0.920i)T \) |
| 5 | \( 1 + (-0.394 - 2.20i)T \) |
| 19 | \( 1 + (-2.35 - 3.66i)T \) |
| good | 3 | \( 1 + (-0.783 + 0.152i)T + (2.78 - 1.12i)T^{2} \) |
| 7 | \( 1 + (-0.314 + 1.17i)T + (-6.06 - 3.5i)T^{2} \) |
| 11 | \( 1 + (4.55 + 0.969i)T + (10.0 + 4.47i)T^{2} \) |
| 13 | \( 1 + (1.74 - 0.214i)T + (12.6 - 3.14i)T^{2} \) |
| 17 | \( 1 + (2.38 - 3.96i)T + (-7.98 - 15.0i)T^{2} \) |
| 23 | \( 1 + (0.680 - 2.22i)T + (-19.0 - 12.8i)T^{2} \) |
| 29 | \( 1 + (-0.663 + 2.65i)T + (-25.6 - 13.6i)T^{2} \) |
| 31 | \( 1 + (4.97 + 0.522i)T + (30.3 + 6.44i)T^{2} \) |
| 37 | \( 1 + (-2.41 - 4.73i)T + (-21.7 + 29.9i)T^{2} \) |
| 41 | \( 1 + (-10.8 + 0.379i)T + (40.9 - 2.86i)T^{2} \) |
| 43 | \( 1 + (2.15 + 3.08i)T + (-14.7 + 40.4i)T^{2} \) |
| 47 | \( 1 + (-5.85 + 3.51i)T + (22.0 - 41.4i)T^{2} \) |
| 53 | \( 1 + (-0.227 - 13.0i)T + (-52.9 + 1.84i)T^{2} \) |
| 59 | \( 1 + (-7.62 + 4.76i)T + (25.8 - 53.0i)T^{2} \) |
| 61 | \( 1 + (8.13 - 4.32i)T + (34.1 - 50.5i)T^{2} \) |
| 67 | \( 1 + (-8.17 + 9.40i)T + (-9.32 - 66.3i)T^{2} \) |
| 71 | \( 1 + (6.58 - 3.21i)T + (43.7 - 55.9i)T^{2} \) |
| 73 | \( 1 + (15.4 + 1.89i)T + (70.8 + 17.6i)T^{2} \) |
| 79 | \( 1 + (0.0844 - 0.125i)T + (-29.5 - 73.2i)T^{2} \) |
| 83 | \( 1 + (-0.317 - 0.392i)T + (-17.2 + 81.1i)T^{2} \) |
| 89 | \( 1 + (0.449 - 12.8i)T + (-88.7 - 6.20i)T^{2} \) |
| 97 | \( 1 + (-6.82 + 5.93i)T + (13.4 - 96.0i)T^{2} \) |
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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
−10.52552498391972347469092670691, −9.587297060870319247313652748969, −8.514716681361984625061614229998, −7.68995437412328613302715214736, −7.35618049550645982321102599834, −6.01495107925540638600290921704, −5.55448196280644555177750795470, −4.19473948910343568126056075290, −3.14699513908145888636343135094, −2.26566980737597608587713073100,
0.40985502242590808493661926524, 2.27262962985968642142448689704, 2.85559607434208241494462821873, 4.31565729857417960653495762389, 5.17667470906611213504895112101, 5.71145393883571034789971712671, 7.25479151303321905382315841637, 8.196977338741525952580753044770, 9.064860622232990584168270858521, 9.411410172134789682115003749610
