L(s) = 1 | + (1.52 − 2.63i)2-s + (−0.626 − 1.08i)4-s + (−2.5 − 4.33i)5-s + (6.85 − 11.8i)7-s + 20.5·8-s − 15.2·10-s + (15.9 − 27.5i)11-s + (−29.1 − 50.4i)13-s + (−20.8 − 36.1i)14-s + (36.2 − 62.7i)16-s − 109.·17-s + 129.·19-s + (−3.13 + 5.42i)20-s + (−48.3 − 83.7i)22-s + (39.8 + 68.9i)23-s + ⋯ |
L(s) = 1 | + (0.537 − 0.931i)2-s + (−0.0782 − 0.135i)4-s + (−0.223 − 0.387i)5-s + (0.370 − 0.641i)7-s + 0.907·8-s − 0.480·10-s + (0.435 − 0.755i)11-s + (−0.621 − 1.07i)13-s + (−0.398 − 0.689i)14-s + (0.566 − 0.980i)16-s − 1.55·17-s + 1.56·19-s + (−0.0349 + 0.0606i)20-s + (−0.468 − 0.811i)22-s + (0.361 + 0.625i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.331 + 0.943i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.331 + 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.31048 - 1.84882i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.31048 - 1.84882i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (2.5 + 4.33i)T \) |
good | 2 | \( 1 + (-1.52 + 2.63i)T + (-4 - 6.92i)T^{2} \) |
| 7 | \( 1 + (-6.85 + 11.8i)T + (-171.5 - 297. i)T^{2} \) |
| 11 | \( 1 + (-15.9 + 27.5i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (29.1 + 50.4i)T + (-1.09e3 + 1.90e3i)T^{2} \) |
| 17 | \( 1 + 109.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 129.T + 6.85e3T^{2} \) |
| 23 | \( 1 + (-39.8 - 68.9i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-4.51 + 7.82i)T + (-1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + (-16.6 - 28.9i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + 22.1T + 5.06e4T^{2} \) |
| 41 | \( 1 + (-60.8 - 105. i)T + (-3.44e4 + 5.96e4i)T^{2} \) |
| 43 | \( 1 + (5.07 - 8.78i)T + (-3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 + (220. - 382. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 - 593.T + 1.48e5T^{2} \) |
| 59 | \( 1 + (-221. - 383. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (72.2 - 125. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (431. + 747. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 - 818.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 495.T + 3.89e5T^{2} \) |
| 79 | \( 1 + (585. - 1.01e3i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + (212. - 367. i)T + (-2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 + 1.03e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + (799. - 1.38e3i)T + (-4.56e5 - 7.90e5i)T^{2} \) |
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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
−12.39246352101960394427590608530, −11.42764108297632962321734385528, −10.81314521781486632317543623996, −9.553309790740123986534445715192, −8.136796526399458649844563282299, −7.15588126069327414273215161245, −5.28974645685562749554359310103, −4.14652166103593380088464071650, −2.91885572122832416734816677724, −1.06784922340211357697545576412,
2.10071213109022541775525048691, 4.27592005843504404333362605687, 5.26339699237645080961616457698, 6.68703711938294539640677329511, 7.24457492924593049884275531980, 8.701372653000478065194811955498, 9.882368245621858647574028214544, 11.24222957230136535278606457256, 12.03141439613498649685673704145, 13.39196098356653859311295676855