| L(s) = 1 | + (0.163 − 0.928i)2-s + (6.68 + 2.43i)4-s + (3.55 − 1.29i)5-s + (−11.7 − 20.2i)7-s + (7.12 − 12.3i)8-s + (−0.620 − 3.51i)10-s + (8.50 − 14.7i)11-s + (3.66 − 3.07i)13-s + (−20.7 + 7.55i)14-s + (33.2 + 27.9i)16-s + (9.73 − 55.2i)17-s + (80.5 − 19.1i)19-s + 26.9·20-s + (−12.2 − 10.3i)22-s + (83.8 + 30.5i)23-s + ⋯ |
| L(s) = 1 | + (0.0579 − 0.328i)2-s + (0.835 + 0.303i)4-s + (0.318 − 0.115i)5-s + (−0.632 − 1.09i)7-s + (0.314 − 0.545i)8-s + (−0.0196 − 0.111i)10-s + (0.233 − 0.403i)11-s + (0.0780 − 0.0655i)13-s + (−0.396 + 0.144i)14-s + (0.519 + 0.436i)16-s + (0.138 − 0.787i)17-s + (0.972 − 0.231i)19-s + 0.300·20-s + (−0.119 − 0.0998i)22-s + (0.759 + 0.276i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 + 0.924i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.382 + 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.78169 - 1.19093i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.78169 - 1.19093i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 19 | \( 1 + (-80.5 + 19.1i)T \) |
| good | 2 | \( 1 + (-0.163 + 0.928i)T + (-7.51 - 2.73i)T^{2} \) |
| 5 | \( 1 + (-3.55 + 1.29i)T + (95.7 - 80.3i)T^{2} \) |
| 7 | \( 1 + (11.7 + 20.2i)T + (-171.5 + 297. i)T^{2} \) |
| 11 | \( 1 + (-8.50 + 14.7i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-3.66 + 3.07i)T + (381. - 2.16e3i)T^{2} \) |
| 17 | \( 1 + (-9.73 + 55.2i)T + (-4.61e3 - 1.68e3i)T^{2} \) |
| 23 | \( 1 + (-83.8 - 30.5i)T + (9.32e3 + 7.82e3i)T^{2} \) |
| 29 | \( 1 + (29.8 + 169. i)T + (-2.29e4 + 8.34e3i)T^{2} \) |
| 31 | \( 1 + (48.1 + 83.4i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 - 339.T + 5.06e4T^{2} \) |
| 41 | \( 1 + (339. + 285. i)T + (1.19e4 + 6.78e4i)T^{2} \) |
| 43 | \( 1 + (253. - 92.2i)T + (6.09e4 - 5.11e4i)T^{2} \) |
| 47 | \( 1 + (-84.2 - 477. i)T + (-9.75e4 + 3.55e4i)T^{2} \) |
| 53 | \( 1 + (-594. - 216. i)T + (1.14e5 + 9.56e4i)T^{2} \) |
| 59 | \( 1 + (114. - 648. i)T + (-1.92e5 - 7.02e4i)T^{2} \) |
| 61 | \( 1 + (135. + 49.4i)T + (1.73e5 + 1.45e5i)T^{2} \) |
| 67 | \( 1 + (-62.4 - 354. i)T + (-2.82e5 + 1.02e5i)T^{2} \) |
| 71 | \( 1 + (998. - 363. i)T + (2.74e5 - 2.30e5i)T^{2} \) |
| 73 | \( 1 + (-227. - 190. i)T + (6.75e4 + 3.83e5i)T^{2} \) |
| 79 | \( 1 + (81.3 + 68.2i)T + (8.56e4 + 4.85e5i)T^{2} \) |
| 83 | \( 1 + (-332. - 576. i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 + (-252. + 212. i)T + (1.22e5 - 6.94e5i)T^{2} \) |
| 97 | \( 1 + (-4.50 + 25.5i)T + (-8.57e5 - 3.12e5i)T^{2} \) |
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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
−11.92515032547191496560858673249, −11.22462874627139673109371385705, −10.17617615544673613242710356309, −9.347957274411430541921302238843, −7.67702411641196723163826917971, −6.97627875696827011880789828386, −5.76082742102701714031597852755, −3.97725835279006234384795085609, −2.85585205287682643208205945940, −1.03107490900307630627147872882,
1.81546915213480003716498102037, 3.15395224542475159577644612069, 5.22687867067008104185407327249, 6.14681658515571889786331293598, 7.00688812967181355194121750939, 8.346170091985701023311548211100, 9.536392462088464868377776405228, 10.41762521282003528689357251160, 11.63024511131903079882713381854, 12.33945031069794024691570574731
