| L(s) = 1 | + (0.642 − 1.26i)2-s + (−0.270 + 1.71i)3-s + (−1.17 − 1.61i)4-s + (−2.43 + 4.36i)5-s + (1.98 + 1.43i)6-s + (4.46 + 4.46i)7-s + (−2.79 + 0.442i)8-s + (−2.85 − 0.927i)9-s + (3.93 + 5.87i)10-s + (4.11 + 12.6i)11-s + (3.08 − 1.57i)12-s + (3.12 + 6.13i)13-s + (8.50 − 2.76i)14-s + (−6.80 − 5.35i)15-s + (−1.23 + 3.80i)16-s + (1.26 + 7.98i)17-s + ⋯ |
| L(s) = 1 | + (0.321 − 0.630i)2-s + (−0.0903 + 0.570i)3-s + (−0.293 − 0.404i)4-s + (−0.487 + 0.872i)5-s + (0.330 + 0.239i)6-s + (0.638 + 0.638i)7-s + (−0.349 + 0.0553i)8-s + (−0.317 − 0.103i)9-s + (0.393 + 0.587i)10-s + (0.374 + 1.15i)11-s + (0.257 − 0.131i)12-s + (0.240 + 0.472i)13-s + (0.607 − 0.197i)14-s + (−0.453 − 0.357i)15-s + (−0.0772 + 0.237i)16-s + (0.0743 + 0.469i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.599 - 0.800i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.599 - 0.800i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.30618 + 0.654047i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.30618 + 0.654047i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.642 + 1.26i)T \) |
| 3 | \( 1 + (0.270 - 1.71i)T \) |
| 5 | \( 1 + (2.43 - 4.36i)T \) |
| good | 7 | \( 1 + (-4.46 - 4.46i)T + 49iT^{2} \) |
| 11 | \( 1 + (-4.11 - 12.6i)T + (-97.8 + 71.1i)T^{2} \) |
| 13 | \( 1 + (-3.12 - 6.13i)T + (-99.3 + 136. i)T^{2} \) |
| 17 | \( 1 + (-1.26 - 7.98i)T + (-274. + 89.3i)T^{2} \) |
| 19 | \( 1 + (-6.62 + 9.11i)T + (-111. - 343. i)T^{2} \) |
| 23 | \( 1 + (-0.409 - 0.208i)T + (310. + 427. i)T^{2} \) |
| 29 | \( 1 + (2.55 + 3.51i)T + (-259. + 799. i)T^{2} \) |
| 31 | \( 1 + (22.7 + 16.5i)T + (296. + 913. i)T^{2} \) |
| 37 | \( 1 + (-54.4 + 27.7i)T + (804. - 1.10e3i)T^{2} \) |
| 41 | \( 1 + (19.2 - 59.3i)T + (-1.35e3 - 988. i)T^{2} \) |
| 43 | \( 1 + (-23.6 + 23.6i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (9.41 + 1.49i)T + (2.10e3 + 682. i)T^{2} \) |
| 53 | \( 1 + (-6.51 + 41.1i)T + (-2.67e3 - 868. i)T^{2} \) |
| 59 | \( 1 + (76.9 + 25.0i)T + (2.81e3 + 2.04e3i)T^{2} \) |
| 61 | \( 1 + (-29.1 - 89.7i)T + (-3.01e3 + 2.18e3i)T^{2} \) |
| 67 | \( 1 + (13.6 + 86.2i)T + (-4.26e3 + 1.38e3i)T^{2} \) |
| 71 | \( 1 + (-26.9 + 19.5i)T + (1.55e3 - 4.79e3i)T^{2} \) |
| 73 | \( 1 + (-104. - 53.1i)T + (3.13e3 + 4.31e3i)T^{2} \) |
| 79 | \( 1 + (-15.5 - 21.3i)T + (-1.92e3 + 5.93e3i)T^{2} \) |
| 83 | \( 1 + (-152. + 24.2i)T + (6.55e3 - 2.12e3i)T^{2} \) |
| 89 | \( 1 + (52.8 - 17.1i)T + (6.40e3 - 4.65e3i)T^{2} \) |
| 97 | \( 1 + (-72.4 - 11.4i)T + (8.94e3 + 2.90e3i)T^{2} \) |
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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
−12.69745074813416013931470009393, −11.64508926368532086590869439011, −11.17038950905229949709191031071, −10.03248916494672127528403732795, −9.094768675547607678234739026598, −7.69554082419793648276469006016, −6.29410802413649926724612655505, −4.84568194544104411864678766463, −3.77726638102038657161021648595, −2.22437504678530298660972131754,
0.937914691446297528924428474831, 3.59510999170654880028597000806, 4.95187856364735672948087056953, 6.04975854127109337215838202758, 7.48399584374519547337453320062, 8.153620078133654552494090452386, 9.145722520712380798768991392349, 10.90688399686358617132521813713, 11.80530862183759277134382712993, 12.77449024500253193204701947834
