| L(s) = 1 | + (0.623 − 0.109i)2-s + (2.58 + 1.52i)3-s + (−3.38 + 1.23i)4-s + (1.20 + 1.44i)5-s + (1.77 + 0.667i)6-s + (1.68 + 2.91i)7-s + (−4.16 + 2.40i)8-s + (4.34 + 7.88i)9-s + (0.912 + 0.765i)10-s + 3.91i·11-s + (−10.6 − 1.98i)12-s + (−1.33 − 1.12i)13-s + (1.36 + 1.63i)14-s + (0.923 + 5.56i)15-s + (8.69 − 7.29i)16-s + (16.7 + 19.9i)17-s + ⋯ |
| L(s) = 1 | + (0.311 − 0.0549i)2-s + (0.860 + 0.508i)3-s + (−0.845 + 0.307i)4-s + (0.241 + 0.288i)5-s + (0.296 + 0.111i)6-s + (0.240 + 0.416i)7-s + (−0.520 + 0.300i)8-s + (0.482 + 0.875i)9-s + (0.0912 + 0.0765i)10-s + 0.356i·11-s + (−0.884 − 0.165i)12-s + (−0.102 − 0.0863i)13-s + (0.0978 + 0.116i)14-s + (0.0615 + 0.371i)15-s + (0.543 − 0.455i)16-s + (0.985 + 1.17i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.219 - 0.975i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.219 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.49280 + 1.19384i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.49280 + 1.19384i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-2.58 - 1.52i)T \) |
| 19 | \( 1 + (12.3 - 14.4i)T \) |
| good | 2 | \( 1 + (-0.623 + 0.109i)T + (3.75 - 1.36i)T^{2} \) |
| 5 | \( 1 + (-1.20 - 1.44i)T + (-4.34 + 24.6i)T^{2} \) |
| 7 | \( 1 + (-1.68 - 2.91i)T + (-24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 - 3.91iT - 121T^{2} \) |
| 13 | \( 1 + (1.33 + 1.12i)T + (29.3 + 166. i)T^{2} \) |
| 17 | \( 1 + (-16.7 - 19.9i)T + (-50.1 + 284. i)T^{2} \) |
| 23 | \( 1 + (13.9 + 38.3i)T + (-405. + 340. i)T^{2} \) |
| 29 | \( 1 + (4.40 + 12.1i)T + (-644. + 540. i)T^{2} \) |
| 31 | \( 1 - 51.2T + 961T^{2} \) |
| 37 | \( 1 - 42.8T + 1.36e3T^{2} \) |
| 41 | \( 1 + (59.6 - 10.5i)T + (1.57e3 - 574. i)T^{2} \) |
| 43 | \( 1 + (-25.4 - 9.26i)T + (1.41e3 + 1.18e3i)T^{2} \) |
| 47 | \( 1 + (19.1 + 52.6i)T + (-1.69e3 + 1.41e3i)T^{2} \) |
| 53 | \( 1 + (20.2 + 3.57i)T + (2.63e3 + 960. i)T^{2} \) |
| 59 | \( 1 + (-15.7 + 43.1i)T + (-2.66e3 - 2.23e3i)T^{2} \) |
| 61 | \( 1 + (-22.2 - 18.6i)T + (646. + 3.66e3i)T^{2} \) |
| 67 | \( 1 + (-10.7 + 61.1i)T + (-4.21e3 - 1.53e3i)T^{2} \) |
| 71 | \( 1 + (-22.4 + 3.95i)T + (4.73e3 - 1.72e3i)T^{2} \) |
| 73 | \( 1 + (19.3 + 7.03i)T + (4.08e3 + 3.42e3i)T^{2} \) |
| 79 | \( 1 + (-6.34 + 5.32i)T + (1.08e3 - 6.14e3i)T^{2} \) |
| 83 | \( 1 + (78.0 - 45.0i)T + (3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + (-5.79 - 15.9i)T + (-6.06e3 + 5.09e3i)T^{2} \) |
| 97 | \( 1 + (-1.80 - 10.2i)T + (-8.84e3 + 3.21e3i)T^{2} \) |
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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
−12.81251443620533292830567072695, −12.07393288165177252089028659581, −10.33426688779424865737482757160, −9.855971697285417787165170291866, −8.384374165428731047981062645081, −8.171567180785265747793729554023, −6.20394780227462097252598753235, −4.78340966085620148312648195529, −3.81328928575229703018660377280, −2.41216279521656689055909192455,
1.11018254953827381219142243371, 3.15549743955714263342494348378, 4.49226192423191067960721886460, 5.74770230242107135007039023248, 7.19690943054195626571416625990, 8.255210789672017722929247673359, 9.286816374884873301462216036099, 9.917609870403993209327328206725, 11.53570272940406383705337884870, 12.65669322138259308212170810932
