| L(s) = 1 | + (−0.273 + 0.0481i)2-s + (−2.97 − 0.402i)3-s + (−3.68 + 1.34i)4-s + (−6.04 − 7.19i)5-s + (0.831 − 0.0332i)6-s + (5.17 + 8.95i)7-s + (1.90 − 1.09i)8-s + (8.67 + 2.39i)9-s + (1.99 + 1.67i)10-s − 4.84i·11-s + (11.4 − 2.50i)12-s + (8.80 + 7.39i)13-s + (−1.84 − 2.19i)14-s + (15.0 + 23.8i)15-s + (11.5 − 9.69i)16-s + (12.6 + 15.0i)17-s + ⋯ |
| L(s) = 1 | + (−0.136 + 0.0240i)2-s + (−0.990 − 0.134i)3-s + (−0.921 + 0.335i)4-s + (−1.20 − 1.43i)5-s + (0.138 − 0.00553i)6-s + (0.738 + 1.27i)7-s + (0.238 − 0.137i)8-s + (0.963 + 0.265i)9-s + (0.199 + 0.167i)10-s − 0.440i·11-s + (0.958 − 0.208i)12-s + (0.677 + 0.568i)13-s + (−0.131 − 0.157i)14-s + (1.00 + 1.58i)15-s + (0.722 − 0.605i)16-s + (0.743 + 0.885i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.877 - 0.479i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.877 - 0.479i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.613678 + 0.156742i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.613678 + 0.156742i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (2.97 + 0.402i)T \) |
| 19 | \( 1 + (13.4 + 13.4i)T \) |
| good | 2 | \( 1 + (0.273 - 0.0481i)T + (3.75 - 1.36i)T^{2} \) |
| 5 | \( 1 + (6.04 + 7.19i)T + (-4.34 + 24.6i)T^{2} \) |
| 7 | \( 1 + (-5.17 - 8.95i)T + (-24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + 4.84iT - 121T^{2} \) |
| 13 | \( 1 + (-8.80 - 7.39i)T + (29.3 + 166. i)T^{2} \) |
| 17 | \( 1 + (-12.6 - 15.0i)T + (-50.1 + 284. i)T^{2} \) |
| 23 | \( 1 + (3.86 + 10.6i)T + (-405. + 340. i)T^{2} \) |
| 29 | \( 1 + (-10.5 - 28.9i)T + (-644. + 540. i)T^{2} \) |
| 31 | \( 1 - 6.35T + 961T^{2} \) |
| 37 | \( 1 - 39.7T + 1.36e3T^{2} \) |
| 41 | \( 1 + (-35.4 + 6.25i)T + (1.57e3 - 574. i)T^{2} \) |
| 43 | \( 1 + (-48.4 - 17.6i)T + (1.41e3 + 1.18e3i)T^{2} \) |
| 47 | \( 1 + (-11.3 - 31.0i)T + (-1.69e3 + 1.41e3i)T^{2} \) |
| 53 | \( 1 + (10.6 + 1.87i)T + (2.63e3 + 960. i)T^{2} \) |
| 59 | \( 1 + (-7.59 + 20.8i)T + (-2.66e3 - 2.23e3i)T^{2} \) |
| 61 | \( 1 + (-2.09 - 1.75i)T + (646. + 3.66e3i)T^{2} \) |
| 67 | \( 1 + (9.82 - 55.7i)T + (-4.21e3 - 1.53e3i)T^{2} \) |
| 71 | \( 1 + (47.8 - 8.44i)T + (4.73e3 - 1.72e3i)T^{2} \) |
| 73 | \( 1 + (-16.8 - 6.11i)T + (4.08e3 + 3.42e3i)T^{2} \) |
| 79 | \( 1 + (6.88 - 5.78i)T + (1.08e3 - 6.14e3i)T^{2} \) |
| 83 | \( 1 + (-54.8 + 31.6i)T + (3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + (-44.0 - 120. i)T + (-6.06e3 + 5.09e3i)T^{2} \) |
| 97 | \( 1 + (24.1 + 136. i)T + (-8.84e3 + 3.21e3i)T^{2} \) |
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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
−12.51659157861415869857044913204, −11.82337774791176754704901382325, −10.95332326763356556510171168025, −9.175090338184358909488467131463, −8.537222061193914712265790385345, −7.80412330378592954850286090432, −5.87240193064479613493966138523, −4.86204948863168282099531881555, −4.11373659660297011439420792115, −1.02839266380187205389871622374,
0.66565554403039032103112986707, 3.79287714387232082955782616339, 4.49271569494451588375413121833, 6.05274565136008548327810994269, 7.39571071840027994976100978211, 7.932492991700453652338913550112, 9.914440226326900162854209820162, 10.58957951400760136575699908574, 11.17896223590876163078534551517, 12.18216921064175537456012332689
