| L(s) = 1 | + (1.61 − 1.17i)2-s + (3.00 − 9.24i)3-s + (1.23 − 3.80i)4-s + (−6.01 − 18.4i)6-s + 23.3·7-s + (−2.47 − 7.60i)8-s + (−54.6 − 39.7i)9-s + (6.24 − 4.53i)11-s + (−31.4 − 22.8i)12-s + (−31.6 − 23.0i)13-s + (37.8 − 27.4i)14-s + (−12.9 − 9.40i)16-s + (5.78 + 17.7i)17-s − 135.·18-s + (32.5 + 100. i)19-s + ⋯ |
| L(s) = 1 | + (0.572 − 0.415i)2-s + (0.578 − 1.78i)3-s + (0.154 − 0.475i)4-s + (−0.408 − 1.25i)6-s + 1.26·7-s + (−0.109 − 0.336i)8-s + (−2.02 − 1.47i)9-s + (0.171 − 0.124i)11-s + (−0.757 − 0.550i)12-s + (−0.675 − 0.491i)13-s + (0.721 − 0.524i)14-s + (−0.202 − 0.146i)16-s + (0.0824 + 0.253i)17-s − 1.76·18-s + (0.393 + 1.20i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 250 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.926 + 0.377i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 250 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.926 + 0.377i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.604059 - 3.08364i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.604059 - 3.08364i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.61 + 1.17i)T \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + (-3.00 + 9.24i)T + (-21.8 - 15.8i)T^{2} \) |
| 7 | \( 1 - 23.3T + 343T^{2} \) |
| 11 | \( 1 + (-6.24 + 4.53i)T + (411. - 1.26e3i)T^{2} \) |
| 13 | \( 1 + (31.6 + 23.0i)T + (678. + 2.08e3i)T^{2} \) |
| 17 | \( 1 + (-5.78 - 17.7i)T + (-3.97e3 + 2.88e3i)T^{2} \) |
| 19 | \( 1 + (-32.5 - 100. i)T + (-5.54e3 + 4.03e3i)T^{2} \) |
| 23 | \( 1 + (-127. + 92.4i)T + (3.75e3 - 1.15e4i)T^{2} \) |
| 29 | \( 1 + (13.9 - 42.9i)T + (-1.97e4 - 1.43e4i)T^{2} \) |
| 31 | \( 1 + (-42.7 - 131. i)T + (-2.41e4 + 1.75e4i)T^{2} \) |
| 37 | \( 1 + (125. + 91.2i)T + (1.56e4 + 4.81e4i)T^{2} \) |
| 41 | \( 1 + (15.6 + 11.3i)T + (2.12e4 + 6.55e4i)T^{2} \) |
| 43 | \( 1 - 70.2T + 7.95e4T^{2} \) |
| 47 | \( 1 + (139. - 430. i)T + (-8.39e4 - 6.10e4i)T^{2} \) |
| 53 | \( 1 + (14.9 - 45.9i)T + (-1.20e5 - 8.75e4i)T^{2} \) |
| 59 | \( 1 + (-635. - 461. i)T + (6.34e4 + 1.95e5i)T^{2} \) |
| 61 | \( 1 + (-62.9 + 45.7i)T + (7.01e4 - 2.15e5i)T^{2} \) |
| 67 | \( 1 + (242. + 747. i)T + (-2.43e5 + 1.76e5i)T^{2} \) |
| 71 | \( 1 + (-289. + 892. i)T + (-2.89e5 - 2.10e5i)T^{2} \) |
| 73 | \( 1 + (-2.28 + 1.65i)T + (1.20e5 - 3.69e5i)T^{2} \) |
| 79 | \( 1 + (-241. + 744. i)T + (-3.98e5 - 2.89e5i)T^{2} \) |
| 83 | \( 1 + (-169. - 521. i)T + (-4.62e5 + 3.36e5i)T^{2} \) |
| 89 | \( 1 + (-581. + 422. i)T + (2.17e5 - 6.70e5i)T^{2} \) |
| 97 | \( 1 + (-175. + 541. i)T + (-7.38e5 - 5.36e5i)T^{2} \) |
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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
−11.59669375882850514714565929362, −10.56256000907618187923614701737, −8.958445519002401660116192093801, −8.031965433437083236739524289218, −7.30897245473613054878910728991, −6.16751975344377210269853550022, −5.00586655068622913812021345143, −3.22589650600207751304753680562, −2.01900833222789867453948942060, −1.04236653208224314462396115976,
2.50519640889933658816930829415, 3.81258325613193288443755786566, 4.85318722684980699137940106786, 5.24622151760868161024817545830, 7.16206299535398854981017954143, 8.293416255762332703704710552589, 9.132997122429592634568326721305, 10.00117532330684402713633145320, 11.28843381035020644951223063855, 11.56706151376520245915987666943
