| L(s) = 1 | + (−3.23 + 2.35i)2-s + (19.0 + 13.8i)3-s + (4.94 − 15.2i)4-s − 14.3·5-s − 94.3·6-s + (−43.0 + 132. i)7-s + (19.7 + 60.8i)8-s + (96.9 + 298. i)9-s + (46.2 − 33.6i)10-s + (−30.3 + 93.5i)11-s + (305. − 221. i)12-s + (94.0 + 68.3i)13-s + (−172. − 529. i)14-s + (−273. − 198. i)15-s + (−207. − 150. i)16-s + (111. + 343. i)17-s + ⋯ |
| L(s) = 1 | + (−0.572 + 0.415i)2-s + (1.22 + 0.889i)3-s + (0.154 − 0.475i)4-s − 0.255·5-s − 1.07·6-s + (−0.332 + 1.02i)7-s + (0.109 + 0.336i)8-s + (0.399 + 1.22i)9-s + (0.146 − 0.106i)10-s + (−0.0757 + 0.233i)11-s + (0.612 − 0.444i)12-s + (0.154 + 0.112i)13-s + (−0.234 − 0.722i)14-s + (−0.313 − 0.227i)15-s + (−0.202 − 0.146i)16-s + (0.0936 + 0.288i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.787 - 0.616i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.787 - 0.616i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.504055 + 1.46068i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.504055 + 1.46068i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (3.23 - 2.35i)T \) |
| 31 | \( 1 + (-4.48e3 + 2.91e3i)T \) |
| good | 3 | \( 1 + (-19.0 - 13.8i)T + (75.0 + 231. i)T^{2} \) |
| 5 | \( 1 + 14.3T + 3.12e3T^{2} \) |
| 7 | \( 1 + (43.0 - 132. i)T + (-1.35e4 - 9.87e3i)T^{2} \) |
| 11 | \( 1 + (30.3 - 93.5i)T + (-1.30e5 - 9.46e4i)T^{2} \) |
| 13 | \( 1 + (-94.0 - 68.3i)T + (1.14e5 + 3.53e5i)T^{2} \) |
| 17 | \( 1 + (-111. - 343. i)T + (-1.14e6 + 8.34e5i)T^{2} \) |
| 19 | \( 1 + (1.22e3 - 890. i)T + (7.65e5 - 2.35e6i)T^{2} \) |
| 23 | \( 1 + (-485. - 1.49e3i)T + (-5.20e6 + 3.78e6i)T^{2} \) |
| 29 | \( 1 + (2.67e3 - 1.94e3i)T + (6.33e6 - 1.95e7i)T^{2} \) |
| 37 | \( 1 + 1.10e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + (-6.15e3 + 4.47e3i)T + (3.58e7 - 1.10e8i)T^{2} \) |
| 43 | \( 1 + (-1.63e4 + 1.18e4i)T + (4.54e7 - 1.39e8i)T^{2} \) |
| 47 | \( 1 + (-1.13e4 - 8.21e3i)T + (7.08e7 + 2.18e8i)T^{2} \) |
| 53 | \( 1 + (-7.75e3 - 2.38e4i)T + (-3.38e8 + 2.45e8i)T^{2} \) |
| 59 | \( 1 + (-3.54e4 - 2.57e4i)T + (2.20e8 + 6.79e8i)T^{2} \) |
| 61 | \( 1 - 6.17e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 970.T + 1.35e9T^{2} \) |
| 71 | \( 1 + (-921. - 2.83e3i)T + (-1.45e9 + 1.06e9i)T^{2} \) |
| 73 | \( 1 + (-2.30e4 + 7.09e4i)T + (-1.67e9 - 1.21e9i)T^{2} \) |
| 79 | \( 1 + (1.28e4 + 3.95e4i)T + (-2.48e9 + 1.80e9i)T^{2} \) |
| 83 | \( 1 + (-5.86e4 + 4.26e4i)T + (1.21e9 - 3.74e9i)T^{2} \) |
| 89 | \( 1 + (3.21e4 - 9.89e4i)T + (-4.51e9 - 3.28e9i)T^{2} \) |
| 97 | \( 1 + (2.24e4 - 6.91e4i)T + (-6.94e9 - 5.04e9i)T^{2} \) |
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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
−14.89517672076448514563979795282, −13.70805792108217358880628633818, −12.18603160178019436349136658277, −10.58349823383539352581832709344, −9.449958498310647590609978091055, −8.767162812229682831147209848990, −7.68781729113514447562311892674, −5.82950647831554225481522415314, −3.99067324756823869695465698728, −2.36894621114685039390546046983,
0.75920973580208144036585430036, 2.46948228348723954567641537202, 3.86920862856845803133929409786, 6.76730270899780575740403375315, 7.74881612603266347755324039126, 8.670900879476499690368222324880, 9.930142992775568845389820034707, 11.22625142209957019238648670595, 12.70226430061604274160868388057, 13.44058827206533193841620961199
