| L(s) = 1 | + (3.16 + 1.82i)2-s + (−8.54 − 13.0i)3-s + (−9.32 − 16.1i)4-s + (37.1 − 64.3i)5-s + (−3.22 − 56.8i)6-s + (45.3 + 121. i)7-s − 185. i·8-s + (−96.9 + 222. i)9-s + (235. − 135. i)10-s + (34.1 − 19.6i)11-s + (−130. + 259. i)12-s − 589. i·13-s + (−78.5 + 467. i)14-s + (−1.15e3 + 65.5i)15-s + (40.0 − 69.3i)16-s + (618. + 1.07e3i)17-s + ⋯ |
| L(s) = 1 | + (0.559 + 0.323i)2-s + (−0.548 − 0.836i)3-s + (−0.291 − 0.504i)4-s + (0.664 − 1.15i)5-s + (−0.0365 − 0.645i)6-s + (0.349 + 0.936i)7-s − 1.02i·8-s + (−0.398 + 0.916i)9-s + (0.743 − 0.429i)10-s + (0.0850 − 0.0490i)11-s + (−0.262 + 0.520i)12-s − 0.967i·13-s + (−0.107 + 0.637i)14-s + (−1.32 + 0.0752i)15-s + (0.0390 − 0.0676i)16-s + (0.519 + 0.899i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.282 + 0.959i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.282 + 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.26655 - 0.946849i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.26655 - 0.946849i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (8.54 + 13.0i)T \) |
| 7 | \( 1 + (-45.3 - 121. i)T \) |
| good | 2 | \( 1 + (-3.16 - 1.82i)T + (16 + 27.7i)T^{2} \) |
| 5 | \( 1 + (-37.1 + 64.3i)T + (-1.56e3 - 2.70e3i)T^{2} \) |
| 11 | \( 1 + (-34.1 + 19.6i)T + (8.05e4 - 1.39e5i)T^{2} \) |
| 13 | \( 1 + 589. iT - 3.71e5T^{2} \) |
| 17 | \( 1 + (-618. - 1.07e3i)T + (-7.09e5 + 1.22e6i)T^{2} \) |
| 19 | \( 1 + (-2.39e3 - 1.38e3i)T + (1.23e6 + 2.14e6i)T^{2} \) |
| 23 | \( 1 + (401. + 231. i)T + (3.21e6 + 5.57e6i)T^{2} \) |
| 29 | \( 1 + 5.29e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 + (-2.46e3 + 1.42e3i)T + (1.43e7 - 2.47e7i)T^{2} \) |
| 37 | \( 1 + (3.92e3 - 6.79e3i)T + (-3.46e7 - 6.00e7i)T^{2} \) |
| 41 | \( 1 + 1.21e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 5.35e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + (-3.37e3 + 5.85e3i)T + (-1.14e8 - 1.98e8i)T^{2} \) |
| 53 | \( 1 + (1.38e4 - 8.00e3i)T + (2.09e8 - 3.62e8i)T^{2} \) |
| 59 | \( 1 + (-3.87e3 - 6.70e3i)T + (-3.57e8 + 6.19e8i)T^{2} \) |
| 61 | \( 1 + (-4.86e3 - 2.80e3i)T + (4.22e8 + 7.31e8i)T^{2} \) |
| 67 | \( 1 + (-6.44e3 - 1.11e4i)T + (-6.75e8 + 1.16e9i)T^{2} \) |
| 71 | \( 1 - 6.06e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 + (-4.47e4 + 2.58e4i)T + (1.03e9 - 1.79e9i)T^{2} \) |
| 79 | \( 1 + (3.36e4 - 5.83e4i)T + (-1.53e9 - 2.66e9i)T^{2} \) |
| 83 | \( 1 - 8.71e3T + 3.93e9T^{2} \) |
| 89 | \( 1 + (9.05e3 - 1.56e4i)T + (-2.79e9 - 4.83e9i)T^{2} \) |
| 97 | \( 1 - 1.86e4iT - 8.58e9T^{2} \) |
| 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
−16.94237721514900337776876171239, −15.52543384294801181993487391569, −14.01077595910662352037650672611, −12.96698464032128802772850902828, −12.04774301012861597137841154130, −9.921717109172557511327661482533, −8.233914071818272756161603189827, −5.89946988396826368015353617927, −5.25399044727373566650801435130, −1.25867912967878755198415550160,
3.31595406149055639177421931200, 4.97801429565274637155895898228, 7.04100628899368742720684976965, 9.466674609175626346855942408205, 10.84005245061697451934870762987, 11.81426363669865074031935921501, 13.84045091602068254854164331940, 14.35886464142510000797659151639, 16.22173637437033132215570720557, 17.41933071087754997769522427873
