| L(s) = 1 | + (2 − 3.46i)2-s + (4.5 + 7.79i)3-s + (−7.99 − 13.8i)4-s + (2.25 − 3.89i)5-s + 36·6-s − 63.9·8-s + (−40.5 + 70.1i)9-s + (−9.00 − 15.5i)10-s + (58.0 + 100. i)11-s + (72 − 124. i)12-s + 85.4·13-s + 40.5·15-s + (−128 + 221. i)16-s + (16.6 + 28.8i)17-s + (162 + 280. i)18-s + (317. − 550. i)19-s + ⋯ |
| L(s) = 1 | + (0.353 − 0.612i)2-s + (0.288 + 0.499i)3-s + (−0.249 − 0.433i)4-s + (0.0402 − 0.0697i)5-s + 0.408·6-s − 0.353·8-s + (−0.166 + 0.288i)9-s + (−0.0284 − 0.0493i)10-s + (0.144 + 0.250i)11-s + (0.144 − 0.249i)12-s + 0.140·13-s + 0.0465·15-s + (−0.125 + 0.216i)16-s + (0.0139 + 0.0241i)17-s + (0.117 + 0.204i)18-s + (0.201 − 0.349i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.900 - 0.435i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.900 - 0.435i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.484165626\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.484165626\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-2 + 3.46i)T \) |
| 3 | \( 1 + (-4.5 - 7.79i)T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + (-2.25 + 3.89i)T + (-1.56e3 - 2.70e3i)T^{2} \) |
| 11 | \( 1 + (-58.0 - 100. i)T + (-8.05e4 + 1.39e5i)T^{2} \) |
| 13 | \( 1 - 85.4T + 3.71e5T^{2} \) |
| 17 | \( 1 + (-16.6 - 28.8i)T + (-7.09e5 + 1.22e6i)T^{2} \) |
| 19 | \( 1 + (-317. + 550. i)T + (-1.23e6 - 2.14e6i)T^{2} \) |
| 23 | \( 1 + (1.36e3 - 2.36e3i)T + (-3.21e6 - 5.57e6i)T^{2} \) |
| 29 | \( 1 - 5.86e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + (-139. - 241. i)T + (-1.43e7 + 2.47e7i)T^{2} \) |
| 37 | \( 1 + (1.51e3 - 2.63e3i)T + (-3.46e7 - 6.00e7i)T^{2} \) |
| 41 | \( 1 + 819.T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.11e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + (3.70e3 - 6.41e3i)T + (-1.14e8 - 1.98e8i)T^{2} \) |
| 53 | \( 1 + (-6.84e3 - 1.18e4i)T + (-2.09e8 + 3.62e8i)T^{2} \) |
| 59 | \( 1 + (-1.11e4 - 1.93e4i)T + (-3.57e8 + 6.19e8i)T^{2} \) |
| 61 | \( 1 + (6.34e3 - 1.09e4i)T + (-4.22e8 - 7.31e8i)T^{2} \) |
| 67 | \( 1 + (-2.61e4 - 4.52e4i)T + (-6.75e8 + 1.16e9i)T^{2} \) |
| 71 | \( 1 - 6.02e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + (-3.84e4 - 6.66e4i)T + (-1.03e9 + 1.79e9i)T^{2} \) |
| 79 | \( 1 + (-1.67e4 + 2.90e4i)T + (-1.53e9 - 2.66e9i)T^{2} \) |
| 83 | \( 1 + 6.05e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + (-4.60e4 + 7.98e4i)T + (-2.79e9 - 4.83e9i)T^{2} \) |
| 97 | \( 1 + 1.52e5T + 8.58e9T^{2} \) |
| 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
−11.02229594255246187111543619416, −10.08371368363493660137410922651, −9.328099407550582134273686203077, −8.373228169046046437022322483191, −7.08979245573586115360182728680, −5.74366767397527999003523320786, −4.72409667515195608625694652029, −3.68956975617950421620194180398, −2.58413574235771668159232023638, −1.16551238210768898715563918903,
0.64467589223355718018740164248, 2.34006760858001028025383182342, 3.61278462043272959497986302497, 4.84891800199306761638731825166, 6.13166889015983905626727900124, 6.80950400419099369812627654849, 7.998391760376409433992042955477, 8.610730360258950093824357658209, 9.798378957420809606178297380592, 10.92829727864364096354099214595
