L(s) = 1 | + (0.103 − 0.0865i)2-s + (−1.73 + 0.0351i)3-s + (−0.344 + 1.95i)4-s + (0.862 − 2.36i)5-s + (−0.175 + 0.153i)6-s + (1.62 + 2.81i)7-s + (0.268 + 0.464i)8-s + (2.99 − 0.121i)9-s + (−0.116 − 0.319i)10-s + 5.59i·11-s + (0.527 − 3.39i)12-s + (1.44 + 3.97i)13-s + (0.411 + 0.149i)14-s + (−1.41 + 4.13i)15-s + (−3.65 − 1.33i)16-s + (1.48 − 4.06i)17-s + ⋯ |
L(s) = 1 | + (0.0729 − 0.0612i)2-s + (−0.999 + 0.0202i)3-s + (−0.172 + 0.975i)4-s + (0.385 − 1.05i)5-s + (−0.0717 + 0.0626i)6-s + (0.614 + 1.06i)7-s + (0.0948 + 0.164i)8-s + (0.999 − 0.0405i)9-s + (−0.0367 − 0.100i)10-s + 1.68i·11-s + (0.152 − 0.979i)12-s + (0.401 + 1.10i)13-s + (0.110 + 0.0400i)14-s + (−0.364 + 1.06i)15-s + (−0.914 − 0.332i)16-s + (0.359 − 0.986i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.600 - 0.799i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.600 - 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.853076 + 0.426121i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.853076 + 0.426121i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.73 - 0.0351i)T \) |
| 19 | \( 1 + (2.66 + 3.45i)T \) |
good | 2 | \( 1 + (-0.103 + 0.0865i)T + (0.347 - 1.96i)T^{2} \) |
| 5 | \( 1 + (-0.862 + 2.36i)T + (-3.83 - 3.21i)T^{2} \) |
| 7 | \( 1 + (-1.62 - 2.81i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 - 5.59iT - 11T^{2} \) |
| 13 | \( 1 + (-1.44 - 3.97i)T + (-9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (-1.48 + 4.06i)T + (-13.0 - 10.9i)T^{2} \) |
| 23 | \( 1 + (-3.14 - 0.554i)T + (21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-1.06 + 6.05i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + 6.64iT - 31T^{2} \) |
| 37 | \( 1 - 4.71iT - 37T^{2} \) |
| 41 | \( 1 + (4.53 - 3.80i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-1.19 - 6.77i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (0.139 + 0.0246i)T + (44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + (4.63 + 3.88i)T + (9.20 + 52.1i)T^{2} \) |
| 59 | \( 1 + (0.831 + 4.71i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (-4.55 + 1.65i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (2.35 - 2.80i)T + (-11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-4.47 + 3.75i)T + (12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + (1.71 + 9.70i)T + (-68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (-3.30 + 9.08i)T + (-60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (-3.95 + 2.28i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-2.35 + 13.3i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (-2.21 - 2.64i)T + (-16.8 + 95.5i)T^{2} \) |
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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
−12.76455986299687220795803393805, −11.82612965046451795851694221336, −11.52214407614818526084355071885, −9.638682981391082061588235583616, −9.024143523869415339182421596727, −7.73652397837672265756962730377, −6.52564941947752605607136200784, −4.90615195767766992618205294305, −4.59689968263776523109787853019, −2.02910922901957181974887203325,
1.13272943573205675730810335301, 3.68606435196576836901226659154, 5.33595129012376669822313569451, 6.08766825586346717298849434547, 7.01847641538986716075435833438, 8.474538544338503842764112187155, 10.33064115197518451142814316320, 10.66195937893256648997069021151, 11.01068785971589389247590711484, 12.68874182118847228972473107447