| L(s) = 1 | + (−2.94 + 0.520i)2-s + (−0.769 − 0.916i)3-s + (4.67 − 1.70i)4-s + (6.60 + 2.40i)5-s + (2.74 + 2.30i)6-s + (4.29 − 7.43i)7-s + (−2.52 + 1.45i)8-s + (1.31 − 7.45i)9-s + (−20.7 − 3.65i)10-s + (−4.77 − 8.26i)11-s + (−5.15 − 2.97i)12-s + (11.5 − 13.7i)13-s + (−8.79 + 24.1i)14-s + (−2.87 − 7.90i)15-s + (−8.55 + 7.17i)16-s + (2.30 + 13.0i)17-s + ⋯ |
| L(s) = 1 | + (−1.47 + 0.260i)2-s + (−0.256 − 0.305i)3-s + (1.16 − 0.425i)4-s + (1.32 + 0.480i)5-s + (0.457 + 0.384i)6-s + (0.613 − 1.06i)7-s + (−0.315 + 0.182i)8-s + (0.146 − 0.828i)9-s + (−2.07 − 0.365i)10-s + (−0.433 − 0.751i)11-s + (−0.429 − 0.247i)12-s + (0.890 − 1.06i)13-s + (−0.628 + 1.72i)14-s + (−0.191 − 0.527i)15-s + (−0.534 + 0.448i)16-s + (0.135 + 0.767i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.166 + 0.986i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.166 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.686536 - 0.580418i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.686536 - 0.580418i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 \) |
| good | 2 | \( 1 + (2.94 - 0.520i)T + (3.75 - 1.36i)T^{2} \) |
| 3 | \( 1 + (0.769 + 0.916i)T + (-1.56 + 8.86i)T^{2} \) |
| 5 | \( 1 + (-6.60 - 2.40i)T + (19.1 + 16.0i)T^{2} \) |
| 7 | \( 1 + (-4.29 + 7.43i)T + (-24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (4.77 + 8.26i)T + (-60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (-11.5 + 13.7i)T + (-29.3 - 166. i)T^{2} \) |
| 17 | \( 1 + (-2.30 - 13.0i)T + (-271. + 98.8i)T^{2} \) |
| 23 | \( 1 + (13.3 - 4.84i)T + (405. - 340. i)T^{2} \) |
| 29 | \( 1 + (19.3 + 3.40i)T + (790. + 287. i)T^{2} \) |
| 31 | \( 1 + (6.97 + 4.02i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + 32.1iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (3.54 + 4.22i)T + (-291. + 1.65e3i)T^{2} \) |
| 43 | \( 1 + (68.9 + 25.0i)T + (1.41e3 + 1.18e3i)T^{2} \) |
| 47 | \( 1 + (4.20 - 23.8i)T + (-2.07e3 - 755. i)T^{2} \) |
| 53 | \( 1 + (-20.6 - 56.6i)T + (-2.15e3 + 1.80e3i)T^{2} \) |
| 59 | \( 1 + (-37.6 + 6.64i)T + (3.27e3 - 1.19e3i)T^{2} \) |
| 61 | \( 1 + (-74.3 + 27.0i)T + (2.85e3 - 2.39e3i)T^{2} \) |
| 67 | \( 1 + (33.2 + 5.85i)T + (4.21e3 + 1.53e3i)T^{2} \) |
| 71 | \( 1 + (-8.54 + 23.4i)T + (-3.86e3 - 3.24e3i)T^{2} \) |
| 73 | \( 1 + (21.4 - 17.9i)T + (925. - 5.24e3i)T^{2} \) |
| 79 | \( 1 + (-14.7 - 17.5i)T + (-1.08e3 + 6.14e3i)T^{2} \) |
| 83 | \( 1 + (-53.3 + 92.3i)T + (-3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + (-77.1 + 91.9i)T + (-1.37e3 - 7.80e3i)T^{2} \) |
| 97 | \( 1 + (8.08 - 1.42i)T + (8.84e3 - 3.21e3i)T^{2} \) |
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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
−10.51873140158344162892697264444, −10.26294665171480320406038282889, −9.183944825580361914935618291021, −8.217986475213423099701052965713, −7.40203227945321200172258543071, −6.36520242485627861901460051579, −5.69193729663296577780963116961, −3.65741058570560293441588666332, −1.76784455381392630320421235411, −0.70795830781822465163001490832,
1.68668794376530361587487417573, 2.20768679412560533749241485500, 4.79346523564029769108981873268, 5.50775762073853343893199694561, 6.84366310187313027046598523806, 8.143478599121616010008175330097, 8.805547473788398538987377984678, 9.669387125086608810807709841948, 10.13140789377481719199560230413, 11.23166862604591647273599872407
