| L(s) = 1 | + (−0.830 − 1.81i)2-s + (2.87 − 0.845i)3-s + (−2.61 + 3.02i)4-s + (−14.1 − 9.07i)5-s + (−3.92 − 4.53i)6-s + (−1.02 − 7.15i)7-s + (7.67 + 2.25i)8-s + (7.57 − 4.86i)9-s + (−4.77 + 33.2i)10-s + (−14.1 + 30.9i)11-s + (−4.98 + 10.9i)12-s + (−9.07 + 63.1i)13-s + (−12.1 + 7.81i)14-s + (−48.3 − 14.1i)15-s + (−2.27 − 15.8i)16-s + (−73.6 − 85.0i)17-s + ⋯ |
| L(s) = 1 | + (−0.293 − 0.643i)2-s + (0.553 − 0.162i)3-s + (−0.327 + 0.377i)4-s + (−1.26 − 0.812i)5-s + (−0.267 − 0.308i)6-s + (−0.0555 − 0.386i)7-s + (0.339 + 0.0996i)8-s + (0.280 − 0.180i)9-s + (−0.151 + 1.05i)10-s + (−0.387 + 0.849i)11-s + (−0.119 + 0.262i)12-s + (−0.193 + 1.34i)13-s + (−0.232 + 0.149i)14-s + (−0.832 − 0.244i)15-s + (−0.0355 − 0.247i)16-s + (−1.05 − 1.21i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.631 - 0.775i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.631 - 0.775i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.0626063 + 0.131814i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0626063 + 0.131814i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.830 + 1.81i)T \) |
| 3 | \( 1 + (-2.87 + 0.845i)T \) |
| 23 | \( 1 + (72.6 - 83.0i)T \) |
| good | 5 | \( 1 + (14.1 + 9.07i)T + (51.9 + 113. i)T^{2} \) |
| 7 | \( 1 + (1.02 + 7.15i)T + (-329. + 96.6i)T^{2} \) |
| 11 | \( 1 + (14.1 - 30.9i)T + (-871. - 1.00e3i)T^{2} \) |
| 13 | \( 1 + (9.07 - 63.1i)T + (-2.10e3 - 618. i)T^{2} \) |
| 17 | \( 1 + (73.6 + 85.0i)T + (-699. + 4.86e3i)T^{2} \) |
| 19 | \( 1 + (35.4 - 40.9i)T + (-976. - 6.78e3i)T^{2} \) |
| 29 | \( 1 + (97.5 + 112. i)T + (-3.47e3 + 2.41e4i)T^{2} \) |
| 31 | \( 1 + (78.1 + 22.9i)T + (2.50e4 + 1.61e4i)T^{2} \) |
| 37 | \( 1 + (-152. + 97.7i)T + (2.10e4 - 4.60e4i)T^{2} \) |
| 41 | \( 1 + (-17.5 - 11.2i)T + (2.86e4 + 6.26e4i)T^{2} \) |
| 43 | \( 1 + (415. - 122. i)T + (6.68e4 - 4.29e4i)T^{2} \) |
| 47 | \( 1 - 373.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (94.3 + 656. i)T + (-1.42e5 + 4.19e4i)T^{2} \) |
| 59 | \( 1 + (-45.5 + 317. i)T + (-1.97e5 - 5.78e4i)T^{2} \) |
| 61 | \( 1 + (340. + 99.8i)T + (1.90e5 + 1.22e5i)T^{2} \) |
| 67 | \( 1 + (-59.0 - 129. i)T + (-1.96e5 + 2.27e5i)T^{2} \) |
| 71 | \( 1 + (339. + 743. i)T + (-2.34e5 + 2.70e5i)T^{2} \) |
| 73 | \( 1 + (350. - 404. i)T + (-5.53e4 - 3.85e5i)T^{2} \) |
| 79 | \( 1 + (-84.6 + 589. i)T + (-4.73e5 - 1.38e5i)T^{2} \) |
| 83 | \( 1 + (217. - 140. i)T + (2.37e5 - 5.20e5i)T^{2} \) |
| 89 | \( 1 + (865. - 254. i)T + (5.93e5 - 3.81e5i)T^{2} \) |
| 97 | \( 1 + (97.0 + 62.3i)T + (3.79e5 + 8.30e5i)T^{2} \) |
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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
−11.94880502980047688066198495612, −11.32846587138996013541622309314, −9.787376704543108953992649694109, −8.980000442260694485643210395196, −7.891814053951955744520041402585, −7.09222850072806468539380747827, −4.64234092503926692828897036060, −3.89240290275016545786502341185, −2.01452209047360768971730593133, −0.07218821914560746924550709103,
2.87801396579119692999974025122, 4.16294781528691687448669341169, 5.86665509955194374474501786151, 7.16825193235885369355396885908, 8.153848487528190590449064546172, 8.748889560025287357426727521743, 10.44383702359619211040184005158, 10.98032910702979421117419624249, 12.46728844713076020379656017548, 13.49973515125262351870036051887
