| L(s) = 1 | + (−0.347 + 1.96i)2-s + (−2.29 − 1.92i)3-s + (−3.75 − 1.36i)4-s + (−4.17 + 1.51i)5-s + (4.59 − 3.85i)6-s + (2.30 + 3.98i)7-s + (4 − 6.92i)8-s + (1.56 + 8.86i)9-s + (−1.54 − 8.75i)10-s + (28.5 − 49.3i)11-s + (6 + 10.3i)12-s + (34.3 − 28.8i)13-s + (−8.65 + 3.14i)14-s + (12.5 + 4.55i)15-s + (12.2 + 10.2i)16-s + (14.8 − 84.2i)17-s + ⋯ |
| L(s) = 1 | + (−0.122 + 0.696i)2-s + (−0.442 − 0.371i)3-s + (−0.469 − 0.171i)4-s + (−0.373 + 0.135i)5-s + (0.312 − 0.262i)6-s + (0.124 + 0.215i)7-s + (0.176 − 0.306i)8-s + (0.0578 + 0.328i)9-s + (−0.0488 − 0.276i)10-s + (0.781 − 1.35i)11-s + (0.144 + 0.249i)12-s + (0.732 − 0.614i)13-s + (−0.165 + 0.0601i)14-s + (0.215 + 0.0784i)15-s + (0.191 + 0.160i)16-s + (0.211 − 1.20i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.904 + 0.426i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.904 + 0.426i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.09915 - 0.246090i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.09915 - 0.246090i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.347 - 1.96i)T \) |
| 3 | \( 1 + (2.29 + 1.92i)T \) |
| 19 | \( 1 + (-82.5 - 6.25i)T \) |
| good | 5 | \( 1 + (4.17 - 1.51i)T + (95.7 - 80.3i)T^{2} \) |
| 7 | \( 1 + (-2.30 - 3.98i)T + (-171.5 + 297. i)T^{2} \) |
| 11 | \( 1 + (-28.5 + 49.3i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-34.3 + 28.8i)T + (381. - 2.16e3i)T^{2} \) |
| 17 | \( 1 + (-14.8 + 84.2i)T + (-4.61e3 - 1.68e3i)T^{2} \) |
| 23 | \( 1 + (-10.1 - 3.70i)T + (9.32e3 + 7.82e3i)T^{2} \) |
| 29 | \( 1 + (16.7 + 94.7i)T + (-2.29e4 + 8.34e3i)T^{2} \) |
| 31 | \( 1 + (6.77 + 11.7i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 - 360.T + 5.06e4T^{2} \) |
| 41 | \( 1 + (219. + 183. i)T + (1.19e4 + 6.78e4i)T^{2} \) |
| 43 | \( 1 + (249. - 90.6i)T + (6.09e4 - 5.11e4i)T^{2} \) |
| 47 | \( 1 + (26.8 + 152. i)T + (-9.75e4 + 3.55e4i)T^{2} \) |
| 53 | \( 1 + (571. + 208. i)T + (1.14e5 + 9.56e4i)T^{2} \) |
| 59 | \( 1 + (54.8 - 311. i)T + (-1.92e5 - 7.02e4i)T^{2} \) |
| 61 | \( 1 + (-390. - 141. i)T + (1.73e5 + 1.45e5i)T^{2} \) |
| 67 | \( 1 + (-32.7 - 185. i)T + (-2.82e5 + 1.02e5i)T^{2} \) |
| 71 | \( 1 + (-376. + 137. i)T + (2.74e5 - 2.30e5i)T^{2} \) |
| 73 | \( 1 + (428. + 359. i)T + (6.75e4 + 3.83e5i)T^{2} \) |
| 79 | \( 1 + (-661. - 555. i)T + (8.56e4 + 4.85e5i)T^{2} \) |
| 83 | \( 1 + (-404. - 699. i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 + (-622. + 522. i)T + (1.22e5 - 6.94e5i)T^{2} \) |
| 97 | \( 1 + (-40.7 + 231. i)T + (-8.57e5 - 3.12e5i)T^{2} \) |
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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
−13.32121722898203447063379308583, −11.78288226340730906128256067580, −11.22622284691734868700264298015, −9.675205177080191853184521130514, −8.471800919394280284126351674585, −7.49923392018718929692652430121, −6.25890655328283224102692361003, −5.31375663363594934557926465070, −3.48576562262758838593464188787, −0.77615969760081621195283270860,
1.47927230715903015505307267602, 3.73905693214462480501029175627, 4.69056845308318753324682662799, 6.39574467065260347161046422505, 7.85063239070376414574141651329, 9.210897660235205309087801444082, 10.06844139634776535772660740591, 11.20470440589578051952143820625, 11.96997330688660865840575136839, 12.85787053355895203742782199906
