| L(s) = 1 | + (14.6 + 10.6i)2-s + (8.34 − 25.6i)3-s + (62.2 + 191. i)4-s + (425. − 309. i)5-s + (396. − 288. i)6-s + (−184. − 568. i)7-s + (−411. + 1.26e3i)8-s + (−589. − 428. i)9-s + 9.54e3·10-s + (−2.67e3 + 3.51e3i)11-s + 5.43e3·12-s + (6.64e3 + 4.83e3i)13-s + (3.35e3 − 1.03e4i)14-s + (−4.38e3 − 1.35e4i)15-s + (1.28e3 − 934. i)16-s + (−1.85e4 + 1.34e4i)17-s + ⋯ |
| L(s) = 1 | + (1.29 + 0.942i)2-s + (0.178 − 0.549i)3-s + (0.486 + 1.49i)4-s + (1.52 − 1.10i)5-s + (0.749 − 0.544i)6-s + (−0.203 − 0.626i)7-s + (−0.284 + 0.875i)8-s + (−0.269 − 0.195i)9-s + 3.01·10-s + (−0.605 + 0.795i)11-s + 0.908·12-s + (0.839 + 0.609i)13-s + (0.326 − 1.00i)14-s + (−0.335 − 1.03i)15-s + (0.0785 − 0.0570i)16-s + (−0.915 + 0.664i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.950 - 0.310i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.950 - 0.310i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(4.15018 + 0.661370i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.15018 + 0.661370i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-8.34 + 25.6i)T \) |
| 11 | \( 1 + (2.67e3 - 3.51e3i)T \) |
| good | 2 | \( 1 + (-14.6 - 10.6i)T + (39.5 + 121. i)T^{2} \) |
| 5 | \( 1 + (-425. + 309. i)T + (2.41e4 - 7.43e4i)T^{2} \) |
| 7 | \( 1 + (184. + 568. i)T + (-6.66e5 + 4.84e5i)T^{2} \) |
| 13 | \( 1 + (-6.64e3 - 4.83e3i)T + (1.93e7 + 5.96e7i)T^{2} \) |
| 17 | \( 1 + (1.85e4 - 1.34e4i)T + (1.26e8 - 3.90e8i)T^{2} \) |
| 19 | \( 1 + (1.20e4 - 3.69e4i)T + (-7.23e8 - 5.25e8i)T^{2} \) |
| 23 | \( 1 + 1.44e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + (1.26e4 + 3.89e4i)T + (-1.39e10 + 1.01e10i)T^{2} \) |
| 31 | \( 1 + (1.00e5 + 7.33e4i)T + (8.50e9 + 2.61e10i)T^{2} \) |
| 37 | \( 1 + (-1.33e5 - 4.11e5i)T + (-7.68e10 + 5.57e10i)T^{2} \) |
| 41 | \( 1 + (-2.39e4 + 7.36e4i)T + (-1.57e11 - 1.14e11i)T^{2} \) |
| 43 | \( 1 + 8.03e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + (-2.07e5 + 6.39e5i)T + (-4.09e11 - 2.97e11i)T^{2} \) |
| 53 | \( 1 + (8.94e4 + 6.49e4i)T + (3.63e11 + 1.11e12i)T^{2} \) |
| 59 | \( 1 + (-2.77e5 - 8.54e5i)T + (-2.01e12 + 1.46e12i)T^{2} \) |
| 61 | \( 1 + (1.11e6 - 8.06e5i)T + (9.71e11 - 2.98e12i)T^{2} \) |
| 67 | \( 1 - 1.15e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + (2.02e6 - 1.46e6i)T + (2.81e12 - 8.64e12i)T^{2} \) |
| 73 | \( 1 + (5.27e5 + 1.62e6i)T + (-8.93e12 + 6.49e12i)T^{2} \) |
| 79 | \( 1 + (-3.74e6 - 2.72e6i)T + (5.93e12 + 1.82e13i)T^{2} \) |
| 83 | \( 1 + (3.66e6 - 2.66e6i)T + (8.38e12 - 2.58e13i)T^{2} \) |
| 89 | \( 1 - 9.62e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + (1.41e7 + 1.02e7i)T + (2.49e13 + 7.68e13i)T^{2} \) |
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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
−14.96307380059092531358842662115, −13.60716824029719880199365337147, −13.34759542317223705555496087085, −12.38851647992457375017647534455, −10.01684274363229846610879228842, −8.368415443718029921253355171316, −6.67399301940950515820062024804, −5.70476644876985144054233725236, −4.28432985499427589651901489641, −1.76851776239764169148988134608,
2.31948362352882477163465742797, 3.16324159181733610431058751243, 5.25983247478258881482221429231, 6.24680796444374312668760473540, 9.154807259945126234465184971840, 10.60799144236139111361007368562, 11.11947500613647893563594789761, 13.09255158544943817769226588412, 13.66783898473103236722215991254, 14.69945048006481978636563914151
