| 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
−10.10758115365978447870130549247, −9.851673740050581264722595291421, −8.641154729484948290246109663642, −8.312178140821287306419535897270, −7.06547719088844322130162107828, −6.06295067073678848016870379678, −4.98932715712651215403956551624, −4.23617126608256409864872952682, −3.13935319422009347919358014253, −2.10652928295357249037891456020,
0.48423419457451312943848376773, 2.03560407539309922399595060311, 3.16841736366009613951857425228, 4.12770985531001722332116113289, 5.68181657845540988554346911806, 6.24716414158953099871283524802, 7.37115485699245443427352667014, 8.083860097319239851745352393006, 8.530406864379196078248642541198, 9.915979938714165338197232453115
