| L(s) = 1 | + (0.748 − 1.56i)3-s + (−0.262 − 0.0462i)5-s + (−0.604 − 1.04i)7-s + (−1.88 − 2.33i)9-s + (−2.03 − 1.17i)11-s + (1.01 − 2.79i)13-s + (−0.268 + 0.375i)15-s + (0.576 − 0.687i)17-s + (−1.97 + 3.88i)19-s + (−2.08 + 0.161i)21-s + (−5.53 + 0.976i)23-s + (−4.63 − 1.68i)25-s + (−5.05 + 1.18i)27-s + (1.92 − 1.61i)29-s + (8.98 − 5.18i)31-s + ⋯ |
| L(s) = 1 | + (0.431 − 0.901i)3-s + (−0.117 − 0.0206i)5-s + (−0.228 − 0.395i)7-s + (−0.626 − 0.779i)9-s + (−0.613 − 0.354i)11-s + (0.282 − 0.775i)13-s + (−0.0693 + 0.0968i)15-s + (0.139 − 0.166i)17-s + (−0.454 + 0.890i)19-s + (−0.455 + 0.0351i)21-s + (−1.15 + 0.203i)23-s + (−0.926 − 0.337i)25-s + (−0.973 + 0.228i)27-s + (0.357 − 0.299i)29-s + (1.61 − 0.931i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.887 + 0.460i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.887 + 0.460i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.270556 - 1.10846i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.270556 - 1.10846i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.748 + 1.56i)T \) |
| 19 | \( 1 + (1.97 - 3.88i)T \) |
| good | 5 | \( 1 + (0.262 + 0.0462i)T + (4.69 + 1.71i)T^{2} \) |
| 7 | \( 1 + (0.604 + 1.04i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (2.03 + 1.17i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.01 + 2.79i)T + (-9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (-0.576 + 0.687i)T + (-2.95 - 16.7i)T^{2} \) |
| 23 | \( 1 + (5.53 - 0.976i)T + (21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (-1.92 + 1.61i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (-8.98 + 5.18i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 3.95iT - 37T^{2} \) |
| 41 | \( 1 + (10.4 - 3.79i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (-0.834 + 4.73i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (1.24 + 1.48i)T + (-8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + (-0.998 - 5.66i)T + (-49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (-9.78 - 8.20i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (0.153 + 0.871i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (3.28 + 3.91i)T + (-11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (-1.64 + 9.33i)T + (-66.7 - 24.2i)T^{2} \) |
| 73 | \( 1 + (-0.320 + 0.116i)T + (55.9 - 46.9i)T^{2} \) |
| 79 | \( 1 + (2.33 + 6.42i)T + (-60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (12.2 - 7.05i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (11.5 + 4.20i)T + (68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (-3.88 + 4.62i)T + (-16.8 - 95.5i)T^{2} \) |
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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
−9.915979938714165338197232453115, −8.530406864379196078248642541198, −8.083860097319239851745352393006, −7.37115485699245443427352667014, −6.24716414158953099871283524802, −5.68181657845540988554346911806, −4.12770985531001722332116113289, −3.16841736366009613951857425228, −2.03560407539309922399595060311, −0.48423419457451312943848376773,
2.10652928295357249037891456020, 3.13935319422009347919358014253, 4.23617126608256409864872952682, 4.98932715712651215403956551624, 6.06295067073678848016870379678, 7.06547719088844322130162107828, 8.312178140821287306419535897270, 8.641154729484948290246109663642, 9.851673740050581264722595291421, 10.10758115365978447870130549247
