| L(s) = 1 | + (0.135 + 1.40i)2-s + (−0.881 + 0.471i)3-s + (−1.96 + 0.381i)4-s + (0.479 + 0.584i)5-s + (−0.782 − 1.17i)6-s + (−1.19 + 0.799i)7-s + (−0.802 − 2.71i)8-s + (0.555 − 0.831i)9-s + (−0.757 + 0.754i)10-s + (−5.07 + 1.53i)11-s + (1.55 − 1.26i)12-s + (−2.10 − 1.73i)13-s + (−1.28 − 1.57i)14-s + (−0.698 − 0.289i)15-s + (3.70 − 1.49i)16-s + (−1.24 + 0.516i)17-s + ⋯ |
| L(s) = 1 | + (0.0957 + 0.995i)2-s + (−0.509 + 0.272i)3-s + (−0.981 + 0.190i)4-s + (0.214 + 0.261i)5-s + (−0.319 − 0.480i)6-s + (−0.452 + 0.302i)7-s + (−0.283 − 0.958i)8-s + (0.185 − 0.277i)9-s + (−0.239 + 0.238i)10-s + (−1.52 + 0.463i)11-s + (0.447 − 0.364i)12-s + (−0.584 − 0.479i)13-s + (−0.344 − 0.421i)14-s + (−0.180 − 0.0746i)15-s + (0.927 − 0.374i)16-s + (−0.302 + 0.125i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.527 + 0.849i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.527 + 0.849i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.144542 - 0.260031i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.144542 - 0.260031i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.135 - 1.40i)T \) |
| 3 | \( 1 + (0.881 - 0.471i)T \) |
| good | 5 | \( 1 + (-0.479 - 0.584i)T + (-0.975 + 4.90i)T^{2} \) |
| 7 | \( 1 + (1.19 - 0.799i)T + (2.67 - 6.46i)T^{2} \) |
| 11 | \( 1 + (5.07 - 1.53i)T + (9.14 - 6.11i)T^{2} \) |
| 13 | \( 1 + (2.10 + 1.73i)T + (2.53 + 12.7i)T^{2} \) |
| 17 | \( 1 + (1.24 - 0.516i)T + (12.0 - 12.0i)T^{2} \) |
| 19 | \( 1 + (-0.153 - 1.55i)T + (-18.6 + 3.70i)T^{2} \) |
| 23 | \( 1 + (0.945 - 0.188i)T + (21.2 - 8.80i)T^{2} \) |
| 29 | \( 1 + (0.526 - 1.73i)T + (-24.1 - 16.1i)T^{2} \) |
| 31 | \( 1 + (3.31 + 3.31i)T + 31iT^{2} \) |
| 37 | \( 1 + (3.16 + 0.311i)T + (36.2 + 7.21i)T^{2} \) |
| 41 | \( 1 + (0.0889 + 0.446i)T + (-37.8 + 15.6i)T^{2} \) |
| 43 | \( 1 + (3.49 + 1.86i)T + (23.8 + 35.7i)T^{2} \) |
| 47 | \( 1 + (-4.80 - 11.5i)T + (-33.2 + 33.2i)T^{2} \) |
| 53 | \( 1 + (-2.23 - 7.37i)T + (-44.0 + 29.4i)T^{2} \) |
| 59 | \( 1 + (9.65 - 7.92i)T + (11.5 - 57.8i)T^{2} \) |
| 61 | \( 1 + (-1.18 - 2.22i)T + (-33.8 + 50.7i)T^{2} \) |
| 67 | \( 1 + (1.46 + 2.74i)T + (-37.2 + 55.7i)T^{2} \) |
| 71 | \( 1 + (-2.89 - 4.33i)T + (-27.1 + 65.5i)T^{2} \) |
| 73 | \( 1 + (-2.34 - 1.56i)T + (27.9 + 67.4i)T^{2} \) |
| 79 | \( 1 + (2.08 - 5.03i)T + (-55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + (8.13 - 0.801i)T + (81.4 - 16.1i)T^{2} \) |
| 89 | \( 1 + (-13.4 - 2.66i)T + (82.2 + 34.0i)T^{2} \) |
| 97 | \( 1 + (7.32 + 7.32i)T + 97iT^{2} \) |
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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
−12.30258902073424373198065767550, −10.71348582286384522893440494555, −10.08593181923193895734342365887, −9.209447685817768033631747753465, −8.001654884409527317853928731992, −7.22305041164779641685255504339, −6.09070828852596367021508303436, −5.38381845754450637116599746804, −4.38014087579552595619649215968, −2.81590240039032604685014342211,
0.19340557315791964933486646226, 2.06214841827260008729813928024, 3.37148165065501727922754522469, 4.85511848008367619627115454781, 5.51065791397335433659691208724, 6.90898619047568654014247588678, 8.078178699157420525020754473227, 9.148980378069231948030814989010, 10.08581009308698281358342982465, 10.76749503923070494504559672255
