| L(s) = 1 | + (−0.366 + 1.36i)2-s + (−0.866 − 1.5i)3-s + (−1.73 − i)4-s + (4.41 + 4.41i)5-s + (2.36 − 0.633i)6-s + (2.11 + 7.89i)7-s + (2 − 1.99i)8-s + (−1.5 + 2.59i)9-s + (−7.64 + 4.41i)10-s + (18.5 + 4.96i)11-s + 3.46i·12-s + (−12.9 − 1.42i)13-s − 11.5·14-s + (2.79 − 10.4i)15-s + (1.99 + 3.46i)16-s + (−19.9 − 11.5i)17-s + ⋯ |
| L(s) = 1 | + (−0.183 + 0.683i)2-s + (−0.288 − 0.5i)3-s + (−0.433 − 0.250i)4-s + (0.882 + 0.882i)5-s + (0.394 − 0.105i)6-s + (0.302 + 1.12i)7-s + (0.250 − 0.249i)8-s + (−0.166 + 0.288i)9-s + (−0.764 + 0.441i)10-s + (1.68 + 0.451i)11-s + 0.288i·12-s + (−0.993 − 0.109i)13-s − 0.825·14-s + (0.186 − 0.696i)15-s + (0.124 + 0.216i)16-s + (−1.17 − 0.679i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.327 - 0.944i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.327 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.959235 + 0.683017i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.959235 + 0.683017i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.366 - 1.36i)T \) |
| 3 | \( 1 + (0.866 + 1.5i)T \) |
| 13 | \( 1 + (12.9 + 1.42i)T \) |
| good | 5 | \( 1 + (-4.41 - 4.41i)T + 25iT^{2} \) |
| 7 | \( 1 + (-2.11 - 7.89i)T + (-42.4 + 24.5i)T^{2} \) |
| 11 | \( 1 + (-18.5 - 4.96i)T + (104. + 60.5i)T^{2} \) |
| 17 | \( 1 + (19.9 + 11.5i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (-0.417 + 0.111i)T + (312. - 180.5i)T^{2} \) |
| 23 | \( 1 + (-36.4 + 21.0i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (3.25 + 5.64i)T + (-420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (17.8 + 17.8i)T + 961iT^{2} \) |
| 37 | \( 1 + (1.37 + 0.369i)T + (1.18e3 + 684.5i)T^{2} \) |
| 41 | \( 1 + (10.8 - 40.3i)T + (-1.45e3 - 840.5i)T^{2} \) |
| 43 | \( 1 + (51.1 + 29.5i)T + (924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (-15.0 + 15.0i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 - 8.90T + 2.80e3T^{2} \) |
| 59 | \( 1 + (-11.4 - 42.6i)T + (-3.01e3 + 1.74e3i)T^{2} \) |
| 61 | \( 1 + (-44.8 + 77.6i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-10.2 + 38.2i)T + (-3.88e3 - 2.24e3i)T^{2} \) |
| 71 | \( 1 + (-8.56 + 2.29i)T + (4.36e3 - 2.52e3i)T^{2} \) |
| 73 | \( 1 + (5.92 - 5.92i)T - 5.32e3iT^{2} \) |
| 79 | \( 1 - 115.T + 6.24e3T^{2} \) |
| 83 | \( 1 + (34.2 + 34.2i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 + (-58.0 - 15.5i)T + (6.85e3 + 3.96e3i)T^{2} \) |
| 97 | \( 1 + (129. - 34.6i)T + (8.14e3 - 4.70e3i)T^{2} \) |
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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
−14.70421119198443169599217570972, −13.55290292632246701305069026440, −12.26183401764840487921980219900, −11.19871852399837706516262635719, −9.672501910689072450632794678935, −8.814811929872529475379449746033, −7.01263768164125617316595385394, −6.41204933831995675619945779618, −5.01986733385761207909467476060, −2.26833808309964379848734156608,
1.34172481331503456514989208179, 3.93440274553145140442271240797, 5.11250641520283181732055064418, 6.87258450767649471244018423052, 8.842022789598711718916878496088, 9.493986630827335430761874838348, 10.68289343287142191293046141659, 11.62723073396572357982864676245, 12.90138915821490620126516335883, 13.80009031915004076243281517700
