| L(s) = 1 | + (−1.59 + 0.672i)3-s + (−0.882 − 2.42i)5-s + (−0.818 − 4.64i)7-s + (2.09 − 2.14i)9-s + (0.647 − 1.77i)11-s + (−2.09 + 2.50i)13-s + (3.03 + 3.27i)15-s + (−0.567 + 0.982i)17-s + (−4.48 + 2.58i)19-s + (4.43 + 6.86i)21-s + (−0.560 + 3.18i)23-s + (−1.26 + 1.06i)25-s + (−1.90 + 4.83i)27-s + (3.21 + 3.83i)29-s + (0.775 − 4.40i)31-s + ⋯ |
| L(s) = 1 | + (−0.921 + 0.388i)3-s + (−0.394 − 1.08i)5-s + (−0.309 − 1.75i)7-s + (0.698 − 0.715i)9-s + (0.195 − 0.536i)11-s + (−0.582 + 0.694i)13-s + (0.784 + 0.845i)15-s + (−0.137 + 0.238i)17-s + (−1.02 + 0.594i)19-s + (0.966 + 1.49i)21-s + (−0.116 + 0.663i)23-s + (−0.253 + 0.212i)25-s + (−0.365 + 0.930i)27-s + (0.597 + 0.712i)29-s + (0.139 − 0.790i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.959 - 0.282i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.959 - 0.282i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0419482 + 0.290776i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0419482 + 0.290776i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.59 - 0.672i)T \) |
| good | 5 | \( 1 + (0.882 + 2.42i)T + (-3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (0.818 + 4.64i)T + (-6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (-0.647 + 1.77i)T + (-8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (2.09 - 2.50i)T + (-2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (0.567 - 0.982i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (4.48 - 2.58i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.560 - 3.18i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (-3.21 - 3.83i)T + (-5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (-0.775 + 4.40i)T + (-29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (6.20 + 3.58i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (2.91 + 2.44i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-1.90 + 5.24i)T + (-32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (-2.09 - 11.8i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 - 3.54iT - 53T^{2} \) |
| 59 | \( 1 + (4.29 + 11.7i)T + (-45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (3.67 - 0.647i)T + (57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (0.836 - 0.996i)T + (-11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-2.92 + 5.06i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-2.92 - 5.06i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (1.80 - 1.51i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-8.86 - 10.5i)T + (-14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (-0.302 - 0.524i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (15.8 + 5.78i)T + (74.3 + 62.3i)T^{2} \) |
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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
−9.808721233673811784513856538350, −9.013715467092912667574921623427, −7.943085370316515842170168455279, −7.02704661275354260703560799650, −6.26072739891744315412570742213, −5.07163630370452085945122351760, −4.24318717987989801978424392413, −3.75832224141731999572764434850, −1.33836632074154190931044643417, −0.16779177615864693628345672538,
2.18030450851771435630151542794, 2.97154737745951217162646745312, 4.61396910383645046922311184258, 5.47685198656150506024674354162, 6.47729962527932252922038458927, 6.87345240351686194254375842664, 8.003110320325905983049262788197, 8.903013747327421406669053220578, 10.04871540669230872172977544376, 10.62601668693955640109293686006
