| L(s) = 1 | + (−1.17 − 2.57i)2-s + (−4.98 + 1.46i)3-s + (−5.23 + 6.04i)4-s + (7.54 − 11.7i)5-s + (9.62 + 11.1i)6-s + (−12.2 + 1.75i)7-s + (21.7 + 6.37i)8-s + (22.7 − 14.5i)9-s + (−39.0 − 5.61i)10-s + (−144. − 66.1i)11-s + (17.2 − 37.8i)12-s + (−20.3 + 141. i)13-s + (18.8 + 29.3i)14-s + (−20.4 + 69.5i)15-s + (−9.10 − 63.3i)16-s + (−64.5 + 55.9i)17-s + ⋯ |
| L(s) = 1 | + (−0.293 − 0.643i)2-s + (−0.553 + 0.162i)3-s + (−0.327 + 0.377i)4-s + (0.301 − 0.469i)5-s + (0.267 + 0.308i)6-s + (−0.248 + 0.0357i)7-s + (0.339 + 0.0996i)8-s + (0.280 − 0.180i)9-s + (−0.390 − 0.0561i)10-s + (−1.19 − 0.547i)11-s + (0.119 − 0.262i)12-s + (−0.120 + 0.839i)13-s + (0.0961 + 0.149i)14-s + (−0.0907 + 0.309i)15-s + (−0.0355 − 0.247i)16-s + (−0.223 + 0.193i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.782 - 0.622i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.782 - 0.622i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.819928 + 0.286099i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.819928 + 0.286099i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.17 + 2.57i)T \) |
| 3 | \( 1 + (4.98 - 1.46i)T \) |
| 23 | \( 1 + (-515. + 120. i)T \) |
| good | 5 | \( 1 + (-7.54 + 11.7i)T + (-259. - 568. i)T^{2} \) |
| 7 | \( 1 + (12.2 - 1.75i)T + (2.30e3 - 676. i)T^{2} \) |
| 11 | \( 1 + (144. + 66.1i)T + (9.58e3 + 1.10e4i)T^{2} \) |
| 13 | \( 1 + (20.3 - 141. i)T + (-2.74e4 - 8.04e3i)T^{2} \) |
| 17 | \( 1 + (64.5 - 55.9i)T + (1.18e4 - 8.26e4i)T^{2} \) |
| 19 | \( 1 + (-355. - 308. i)T + (1.85e4 + 1.28e5i)T^{2} \) |
| 29 | \( 1 + (-818. - 944. i)T + (-1.00e5 + 7.00e5i)T^{2} \) |
| 31 | \( 1 + (1.00e3 + 295. i)T + (7.76e5 + 4.99e5i)T^{2} \) |
| 37 | \( 1 + (-1.05e3 - 1.63e3i)T + (-7.78e5 + 1.70e6i)T^{2} \) |
| 41 | \( 1 + (-1.21e3 - 782. i)T + (1.17e6 + 2.57e6i)T^{2} \) |
| 43 | \( 1 + (170. + 582. i)T + (-2.87e6 + 1.84e6i)T^{2} \) |
| 47 | \( 1 - 4.02e3T + 4.87e6T^{2} \) |
| 53 | \( 1 + (3.67e3 - 528. i)T + (7.57e6 - 2.22e6i)T^{2} \) |
| 59 | \( 1 + (510. - 3.55e3i)T + (-1.16e7 - 3.41e6i)T^{2} \) |
| 61 | \( 1 + (-146. + 499. i)T + (-1.16e7 - 7.48e6i)T^{2} \) |
| 67 | \( 1 + (-1.14e3 + 522. i)T + (1.31e7 - 1.52e7i)T^{2} \) |
| 71 | \( 1 + (-330. - 723. i)T + (-1.66e7 + 1.92e7i)T^{2} \) |
| 73 | \( 1 + (5.89e3 - 6.80e3i)T + (-4.04e6 - 2.81e7i)T^{2} \) |
| 79 | \( 1 + (8.77e3 + 1.26e3i)T + (3.73e7 + 1.09e7i)T^{2} \) |
| 83 | \( 1 + (3.23e3 + 5.04e3i)T + (-1.97e7 + 4.31e7i)T^{2} \) |
| 89 | \( 1 + (208. + 711. i)T + (-5.27e7 + 3.39e7i)T^{2} \) |
| 97 | \( 1 + (-1.09e3 + 1.70e3i)T + (-3.67e7 - 8.05e7i)T^{2} \) |
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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
−12.57232907606869914367844480833, −11.45200113523587365600931289825, −10.61691014947969102988154008666, −9.616086288635513082805944943294, −8.670058355360111651467619375531, −7.31345703296958885623890071216, −5.76302717623254134020741151291, −4.68692509511462266707214030941, −3.03363683593832317182662382734, −1.22393723282654656694192392538,
0.47401015063491085898137932356, 2.68457430919368317847071834410, 4.84222209039352804194435482829, 5.81463828112962000132668120224, 7.03501639641628490860713638436, 7.80045662582843155998082541781, 9.330834021057298566110700723113, 10.30742682414167878037424514832, 11.09874702545677857140362140711, 12.59177262889039477392580162794
