| L(s) = 1 | + (−1.22 − 1.22i)3-s + 2.44·7-s + 2.99i·9-s − 4.89·11-s − 2i·13-s + 6·17-s − 4.89i·19-s + (−2.99 − 2.99i)21-s + 2.44i·23-s + (3.67 − 3.67i)27-s − 9.79i·31-s + (5.99 + 5.99i)33-s − 2i·37-s + (−2.44 + 2.44i)39-s + 6i·41-s + ⋯ |
| L(s) = 1 | + (−0.707 − 0.707i)3-s + 0.925·7-s + 0.999i·9-s − 1.47·11-s − 0.554i·13-s + 1.45·17-s − 1.12i·19-s + (−0.654 − 0.654i)21-s + 0.510i·23-s + (0.707 − 0.707i)27-s − 1.75i·31-s + (1.04 + 1.04i)33-s − 0.328i·37-s + (−0.392 + 0.392i)39-s + 0.937i·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.316 + 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.316 + 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.080217738\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.080217738\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.22 + 1.22i)T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 2.44T + 7T^{2} \) |
| 11 | \( 1 + 4.89T + 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 + 4.89iT - 19T^{2} \) |
| 23 | \( 1 - 2.44iT - 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + 9.79iT - 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 - 6iT - 41T^{2} \) |
| 43 | \( 1 - 2.44T + 43T^{2} \) |
| 47 | \( 1 + 12.2iT - 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 + 9.79T + 59T^{2} \) |
| 61 | \( 1 - 8T + 61T^{2} \) |
| 67 | \( 1 - 7.34T + 67T^{2} \) |
| 71 | \( 1 + 4.89T + 71T^{2} \) |
| 73 | \( 1 + 14iT - 73T^{2} \) |
| 79 | \( 1 - 4.89iT - 79T^{2} \) |
| 83 | \( 1 + 7.34iT - 83T^{2} \) |
| 89 | \( 1 + 12iT - 89T^{2} \) |
| 97 | \( 1 + 10iT - 97T^{2} \) |
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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
−9.684745341005508719579288884442, −8.294967596583936632103026081637, −7.79831438691005220718335707384, −7.25001126726232362392264026358, −5.94465182253247385241349778986, −5.34134021971849280562829356947, −4.65964101672166786085914853699, −3.02754418150256193495200735190, −1.93652270288700147074454678191, −0.54377063936868461743456342907,
1.35688950941579948201508818312, 2.93856843572055286182055090923, 4.05820026651018439650738125514, 5.08141218847295693075234329865, 5.46293762174267451844887393495, 6.53826806688443484105150056407, 7.70099128026319700180002899415, 8.254352275886005676073591755565, 9.334402142277235887891941927201, 10.23792399073936791352990384661
