L(s) = 1 | + (1.28 − 0.595i)2-s + (−1.03 − 1.38i)3-s + (1.29 − 1.52i)4-s − 1.31·5-s + (−2.15 − 1.16i)6-s − 1.36i·7-s + (0.744 − 2.72i)8-s + (−0.855 + 2.87i)9-s + (−1.69 + 0.786i)10-s − 1.33i·11-s + (−3.45 − 0.208i)12-s − 4.58i·13-s + (−0.813 − 1.75i)14-s + (1.36 + 1.83i)15-s + (−0.670 − 3.94i)16-s + 4.76i·17-s + ⋯ |
L(s) = 1 | + (0.906 − 0.421i)2-s + (−0.597 − 0.801i)3-s + (0.645 − 0.764i)4-s − 0.590·5-s + (−0.879 − 0.475i)6-s − 0.516i·7-s + (0.263 − 0.964i)8-s + (−0.285 + 0.958i)9-s + (−0.535 + 0.248i)10-s − 0.403i·11-s + (−0.998 − 0.0602i)12-s − 1.27i·13-s + (−0.217 − 0.468i)14-s + (0.352 + 0.472i)15-s + (−0.167 − 0.985i)16-s + 1.15i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.930 + 0.365i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.930 + 0.365i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.285855 - 1.50868i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.285855 - 1.50868i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.28 + 0.595i)T \) |
| 3 | \( 1 + (1.03 + 1.38i)T \) |
| 23 | \( 1 - T \) |
good | 5 | \( 1 + 1.31T + 5T^{2} \) |
| 7 | \( 1 + 1.36iT - 7T^{2} \) |
| 11 | \( 1 + 1.33iT - 11T^{2} \) |
| 13 | \( 1 + 4.58iT - 13T^{2} \) |
| 17 | \( 1 - 4.76iT - 17T^{2} \) |
| 19 | \( 1 + 6.45T + 19T^{2} \) |
| 29 | \( 1 - 4.15T + 29T^{2} \) |
| 31 | \( 1 + 8.95iT - 31T^{2} \) |
| 37 | \( 1 - 2.96iT - 37T^{2} \) |
| 41 | \( 1 - 6.40iT - 41T^{2} \) |
| 43 | \( 1 - 12.4T + 43T^{2} \) |
| 47 | \( 1 - 4.02T + 47T^{2} \) |
| 53 | \( 1 + 9.40T + 53T^{2} \) |
| 59 | \( 1 + 4.77iT - 59T^{2} \) |
| 61 | \( 1 + 13.3iT - 61T^{2} \) |
| 67 | \( 1 - 7.21T + 67T^{2} \) |
| 71 | \( 1 - 14.4T + 71T^{2} \) |
| 73 | \( 1 - 3.85T + 73T^{2} \) |
| 79 | \( 1 - 0.671iT - 79T^{2} \) |
| 83 | \( 1 + 6.46iT - 83T^{2} \) |
| 89 | \( 1 + 4.72iT - 89T^{2} \) |
| 97 | \( 1 + 5.63T + 97T^{2} \) |
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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
−10.93328395244259074111389657171, −9.973725187803471003735771930738, −8.226624413244876076203744234869, −7.64826396253906484550227396270, −6.40462607748608153446577771772, −5.89857301328092901527886453705, −4.65771141947300993410564616609, −3.67500183241649056755461167164, −2.28369578067786481145936470918, −0.69798356836553942086551119535,
2.49798079926615373693612821671, 3.92976118658744037343185627219, 4.53295626275478699988117619168, 5.47705658431648624351803467847, 6.52222171735018080928216186201, 7.23311828733120051551279908024, 8.589334169960985537595369731341, 9.288688724055689937076291177490, 10.63356231621322048131570067132, 11.29811383842747160133237397702