| L(s) = 1 | + (−0.483 + 1.32i)2-s + (3.64 − 0.643i)3-s + (−1.53 − 1.28i)4-s + (0.297 − 0.249i)5-s + (−0.909 + 5.15i)6-s + (−2.96 + 5.13i)7-s + (2.44 − 1.41i)8-s + (4.43 − 1.61i)9-s + (0.188 + 0.516i)10-s + (−8.13 − 14.0i)11-s + (−6.41 − 3.70i)12-s + (1.76 + 0.311i)13-s + (−5.38 − 6.42i)14-s + (0.925 − 1.10i)15-s + (0.694 + 3.93i)16-s + (−24.3 − 8.86i)17-s + ⋯ |
| L(s) = 1 | + (−0.241 + 0.664i)2-s + (1.21 − 0.214i)3-s + (−0.383 − 0.321i)4-s + (0.0595 − 0.0499i)5-s + (−0.151 + 0.859i)6-s + (−0.423 + 0.733i)7-s + (0.306 − 0.176i)8-s + (0.493 − 0.179i)9-s + (0.0188 + 0.0516i)10-s + (−0.739 − 1.28i)11-s + (−0.534 − 0.308i)12-s + (0.135 + 0.0239i)13-s + (−0.384 − 0.458i)14-s + (0.0617 − 0.0735i)15-s + (0.0434 + 0.246i)16-s + (−1.43 − 0.521i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.831 - 0.556i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.831 - 0.556i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.13166 + 0.343772i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.13166 + 0.343772i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.483 - 1.32i)T \) |
| 19 | \( 1 + (-18.6 + 3.52i)T \) |
| good | 3 | \( 1 + (-3.64 + 0.643i)T + (8.45 - 3.07i)T^{2} \) |
| 5 | \( 1 + (-0.297 + 0.249i)T + (4.34 - 24.6i)T^{2} \) |
| 7 | \( 1 + (2.96 - 5.13i)T + (-24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (8.13 + 14.0i)T + (-60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (-1.76 - 0.311i)T + (158. + 57.8i)T^{2} \) |
| 17 | \( 1 + (24.3 + 8.86i)T + (221. + 185. i)T^{2} \) |
| 23 | \( 1 + (-32.3 - 27.1i)T + (91.8 + 520. i)T^{2} \) |
| 29 | \( 1 + (9.40 + 25.8i)T + (-644. + 540. i)T^{2} \) |
| 31 | \( 1 + (-17.4 - 10.0i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 - 19.4iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (18.6 - 3.28i)T + (1.57e3 - 574. i)T^{2} \) |
| 43 | \( 1 + (-37.5 + 31.5i)T + (321. - 1.82e3i)T^{2} \) |
| 47 | \( 1 + (-52.0 + 18.9i)T + (1.69e3 - 1.41e3i)T^{2} \) |
| 53 | \( 1 + (9.03 - 10.7i)T + (-487. - 2.76e3i)T^{2} \) |
| 59 | \( 1 + (-7.58 + 20.8i)T + (-2.66e3 - 2.23e3i)T^{2} \) |
| 61 | \( 1 + (10.9 + 9.16i)T + (646. + 3.66e3i)T^{2} \) |
| 67 | \( 1 + (-12.7 - 34.9i)T + (-3.43e3 + 2.88e3i)T^{2} \) |
| 71 | \( 1 + (20.3 + 24.2i)T + (-875. + 4.96e3i)T^{2} \) |
| 73 | \( 1 + (12.9 + 73.5i)T + (-5.00e3 + 1.82e3i)T^{2} \) |
| 79 | \( 1 + (147. - 25.9i)T + (5.86e3 - 2.13e3i)T^{2} \) |
| 83 | \( 1 + (-29.0 + 50.3i)T + (-3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + (91.8 + 16.2i)T + (7.44e3 + 2.70e3i)T^{2} \) |
| 97 | \( 1 + (-41.6 + 114. i)T + (-7.20e3 - 6.04e3i)T^{2} \) |
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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
−15.78525405997910918463078768397, −15.29169691406551762025182083368, −13.69902557203189629932493593237, −13.34318155103314447190098052411, −11.27350584447770658295052707273, −9.328677422170765535581586242172, −8.666457538680148529334391443891, −7.35170792656978322911963283638, −5.57934502053813509500401116089, −2.98981930150744549574990999046,
2.61989542224974784592724951003, 4.30416273566394480573284751199, 7.20949566102158210062140149860, 8.611245707188486235144403672355, 9.746382417369182099534102974446, 10.77448026129120451903416131059, 12.63493719742427171293485382009, 13.52286962218429433776518288094, 14.67344041827690735527038794678, 15.80987818710697029796735059134
