| L(s) = 1 | + (1.99 + 0.0489i)2-s + (1.15 − 2.76i)3-s + (3.99 + 0.195i)4-s + (3.75 − 1.36i)5-s + (2.45 − 5.47i)6-s + (−12.2 + 2.15i)7-s + (7.97 + 0.586i)8-s + (−6.31 − 6.41i)9-s + (7.57 − 2.54i)10-s + (−5.05 + 13.8i)11-s + (5.17 − 10.8i)12-s + (8.77 + 7.35i)13-s + (−24.5 + 3.70i)14-s + (0.571 − 11.9i)15-s + (15.9 + 1.56i)16-s + (2.73 − 4.73i)17-s + ⋯ |
| L(s) = 1 | + (0.999 + 0.0244i)2-s + (0.386 − 0.922i)3-s + (0.998 + 0.0489i)4-s + (0.751 − 0.273i)5-s + (0.408 − 0.912i)6-s + (−1.74 + 0.307i)7-s + (0.997 + 0.0733i)8-s + (−0.701 − 0.712i)9-s + (0.757 − 0.254i)10-s + (−0.459 + 1.26i)11-s + (0.431 − 0.902i)12-s + (0.674 + 0.566i)13-s + (−1.75 + 0.264i)14-s + (0.0380 − 0.798i)15-s + (0.995 + 0.0977i)16-s + (0.160 − 0.278i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.791 + 0.610i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.791 + 0.610i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.41682 - 0.824176i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.41682 - 0.824176i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.99 - 0.0489i)T \) |
| 3 | \( 1 + (-1.15 + 2.76i)T \) |
| good | 5 | \( 1 + (-3.75 + 1.36i)T + (19.1 - 16.0i)T^{2} \) |
| 7 | \( 1 + (12.2 - 2.15i)T + (46.0 - 16.7i)T^{2} \) |
| 11 | \( 1 + (5.05 - 13.8i)T + (-92.6 - 77.7i)T^{2} \) |
| 13 | \( 1 + (-8.77 - 7.35i)T + (29.3 + 166. i)T^{2} \) |
| 17 | \( 1 + (-2.73 + 4.73i)T + (-144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-20.0 + 11.5i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (15.3 + 2.70i)T + (497. + 180. i)T^{2} \) |
| 29 | \( 1 + (16.7 - 14.0i)T + (146. - 828. i)T^{2} \) |
| 31 | \( 1 + (-11.7 - 2.07i)T + (903. + 328. i)T^{2} \) |
| 37 | \( 1 + (6.45 - 11.1i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (55.7 + 46.7i)T + (291. + 1.65e3i)T^{2} \) |
| 43 | \( 1 + (-20.3 + 55.8i)T + (-1.41e3 - 1.18e3i)T^{2} \) |
| 47 | \( 1 + (20.2 - 3.56i)T + (2.07e3 - 755. i)T^{2} \) |
| 53 | \( 1 + 16.2T + 2.80e3T^{2} \) |
| 59 | \( 1 + (-6.75 - 18.5i)T + (-2.66e3 + 2.23e3i)T^{2} \) |
| 61 | \( 1 + (-5.25 - 29.8i)T + (-3.49e3 + 1.27e3i)T^{2} \) |
| 67 | \( 1 + (-52.6 + 62.8i)T + (-779. - 4.42e3i)T^{2} \) |
| 71 | \( 1 + (-36.4 - 21.0i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (44.1 + 76.5i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (38.2 + 45.5i)T + (-1.08e3 + 6.14e3i)T^{2} \) |
| 83 | \( 1 + (-1.27 - 1.52i)T + (-1.19e3 + 6.78e3i)T^{2} \) |
| 89 | \( 1 + (-45.8 - 79.3i)T + (-3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-64.3 - 23.4i)T + (7.20e3 + 6.04e3i)T^{2} \) |
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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
−13.40223180029190048636200200494, −12.58948102363242056595958473098, −11.87573204700391462353220172361, −10.08044820572597183068073954533, −9.133164554454540972934379722920, −7.31363457217866228080856103009, −6.52080594052911786224204601179, −5.46358443256171734312119233736, −3.41600243302278181577641854972, −2.07918498831307230202376032507,
2.96817033858245961071259022285, 3.66987185840039127541202360713, 5.63837634163700057998296713712, 6.24597663475088954624452591166, 8.051778798939982048123800656659, 9.763318404496930385106060698119, 10.26657454036240653838285252501, 11.41384499128140355763063797852, 13.01622812009257962787528803531, 13.58321712595749905513028874370
