| L(s) = 1 | + (1.66 − 2.28i)2-s + (0.309 − 0.951i)3-s + (−2.47 − 7.60i)4-s + (35.4 − 25.7i)5-s + (−1.66 − 2.29i)6-s + (−65.6 + 21.3i)7-s + (−21.5 − 6.99i)8-s + (64.7 + 47.0i)9-s − 123. i·10-s + (65.6 + 101. i)11-s − 8.00·12-s + (−100. + 138. i)13-s + (−60.2 + 185. i)14-s + (−13.5 − 41.6i)15-s + (−51.7 + 37.6i)16-s + (−130. − 179. i)17-s + ⋯ |
| L(s) = 1 | + (0.415 − 0.572i)2-s + (0.0343 − 0.105i)3-s + (−0.154 − 0.475i)4-s + (1.41 − 1.02i)5-s + (−0.0462 − 0.0636i)6-s + (−1.33 + 0.435i)7-s + (−0.336 − 0.109i)8-s + (0.799 + 0.580i)9-s − 1.23i·10-s + (0.542 + 0.840i)11-s − 0.0556·12-s + (−0.593 + 0.816i)13-s + (−0.307 + 0.946i)14-s + (−0.0601 − 0.185i)15-s + (−0.202 + 0.146i)16-s + (−0.451 − 0.621i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.482 + 0.875i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.482 + 0.875i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.44924 - 0.855934i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.44924 - 0.855934i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.66 + 2.28i)T \) |
| 11 | \( 1 + (-65.6 - 101. i)T \) |
| good | 3 | \( 1 + (-0.309 + 0.951i)T + (-65.5 - 47.6i)T^{2} \) |
| 5 | \( 1 + (-35.4 + 25.7i)T + (193. - 594. i)T^{2} \) |
| 7 | \( 1 + (65.6 - 21.3i)T + (1.94e3 - 1.41e3i)T^{2} \) |
| 13 | \( 1 + (100. - 138. i)T + (-8.82e3 - 2.71e4i)T^{2} \) |
| 17 | \( 1 + (130. + 179. i)T + (-2.58e4 + 7.94e4i)T^{2} \) |
| 19 | \( 1 + (-141. - 46.0i)T + (1.05e5 + 7.66e4i)T^{2} \) |
| 23 | \( 1 + 272.T + 2.79e5T^{2} \) |
| 29 | \( 1 + (-571. + 185. i)T + (5.72e5 - 4.15e5i)T^{2} \) |
| 31 | \( 1 + (714. + 519. i)T + (2.85e5 + 8.78e5i)T^{2} \) |
| 37 | \( 1 + (-440. - 1.35e3i)T + (-1.51e6 + 1.10e6i)T^{2} \) |
| 41 | \( 1 + (1.05e3 + 341. i)T + (2.28e6 + 1.66e6i)T^{2} \) |
| 43 | \( 1 + 1.48e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 + (772. - 2.37e3i)T + (-3.94e6 - 2.86e6i)T^{2} \) |
| 53 | \( 1 + (3.07e3 + 2.23e3i)T + (2.43e6 + 7.50e6i)T^{2} \) |
| 59 | \( 1 + (338. + 1.04e3i)T + (-9.80e6 + 7.12e6i)T^{2} \) |
| 61 | \( 1 + (1.85e3 + 2.55e3i)T + (-4.27e6 + 1.31e7i)T^{2} \) |
| 67 | \( 1 - 251.T + 2.01e7T^{2} \) |
| 71 | \( 1 + (-3.59e3 + 2.60e3i)T + (7.85e6 - 2.41e7i)T^{2} \) |
| 73 | \( 1 + (-3.98e3 + 1.29e3i)T + (2.29e7 - 1.66e7i)T^{2} \) |
| 79 | \( 1 + (3.10e3 - 4.28e3i)T + (-1.20e7 - 3.70e7i)T^{2} \) |
| 83 | \( 1 + (-261. - 359. i)T + (-1.46e7 + 4.51e7i)T^{2} \) |
| 89 | \( 1 - 8.93e3T + 6.27e7T^{2} \) |
| 97 | \( 1 + (-1.68e3 - 1.22e3i)T + (2.73e7 + 8.41e7i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.99388080848342194042483201105, −15.87988633973428154047969055035, −13.97592255226867294382788437874, −13.03584585339722227968666907614, −12.21878548460562405843594339755, −9.838195204057101836585695918973, −9.410277143611818907374482923093, −6.49995504162733510286857989579, −4.81067997770020638707388062760, −1.99479565140039701177443395087,
3.30570143357465197730561399986, 6.03748520886147936328687265830, 6.90630606187539462479666828803, 9.444841167068283362892238630656, 10.43650317039376635057643579468, 12.74321090773273762349081722514, 13.69506729096731214395024584698, 14.80704018847854191397594123171, 16.13724434617722352569608103807, 17.39462184639280831168757754832
