L(s) = 1 | + (−1.27 − 2.20i)2-s + (−1.07 − 1.86i)3-s + (−2.24 + 3.88i)4-s + (−2.74 + 4.74i)6-s + (−1.46 + 2.54i)7-s + 6.31·8-s + (−0.817 + 1.41i)9-s + (0.317 + 0.550i)11-s + 9.64·12-s + (−0.0716 + 3.60i)13-s + 7.48·14-s + (−3.55 − 6.16i)16-s + (−0.611 + 1.05i)17-s + 4.16·18-s + (−0.682 + 1.18i)19-s + ⋯ |
L(s) = 1 | + (−0.900 − 1.55i)2-s + (−0.621 − 1.07i)3-s + (−1.12 + 1.94i)4-s + (−1.11 + 1.93i)6-s + (−0.555 + 0.961i)7-s + 2.23·8-s + (−0.272 + 0.472i)9-s + (0.0957 + 0.165i)11-s + 2.78·12-s + (−0.0198 + 0.999i)13-s + 1.99·14-s + (−0.889 − 1.54i)16-s + (−0.148 + 0.257i)17-s + 0.981·18-s + (−0.156 + 0.271i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 + 0.271i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.962 + 0.271i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.321511 - 0.0445323i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.321511 - 0.0445323i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 13 | \( 1 + (0.0716 - 3.60i)T \) |
good | 2 | \( 1 + (1.27 + 2.20i)T + (-1 + 1.73i)T^{2} \) |
| 3 | \( 1 + (1.07 + 1.86i)T + (-1.5 + 2.59i)T^{2} \) |
| 7 | \( 1 + (1.46 - 2.54i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.317 - 0.550i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (0.611 - 1.05i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.682 - 1.18i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.07 - 1.86i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (1.5 + 2.59i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 8.96T + 31T^{2} \) |
| 37 | \( 1 + (-0.611 - 1.05i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-4.98 - 8.62i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (0.683 - 1.18i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 6.16T + 47T^{2} \) |
| 53 | \( 1 + 0.642T + 53T^{2} \) |
| 59 | \( 1 + (3.79 - 6.57i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.13 + 1.96i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-4.01 - 6.95i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (1.31 - 2.28i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 - 10.3T + 73T^{2} \) |
| 79 | \( 1 - 1.03T + 79T^{2} \) |
| 83 | \( 1 + 11.8T + 83T^{2} \) |
| 89 | \( 1 + (-6.27 - 10.8i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (7.39 - 12.8i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.57498532207817435327416615293, −10.94939794294647601446255928946, −9.607954370289128789237173991285, −9.145783057831013364512599649215, −8.009285759574108533886892975983, −6.90810412599483757226532316061, −5.78891947910469262592889551719, −3.98468387414823260351050599000, −2.51296369904954113382691381509, −1.47798420532240754271086828667,
0.35153418649211656866614430062, 3.82628270283402294002025052495, 5.05285195064736343337465957630, 5.80924403198347602223039898062, 6.93624391417427160253158638101, 7.67851550299640979567448072412, 8.921837887794307421561953639868, 9.644446408482332870868481937290, 10.52584730786406572976888174895, 10.92170075198947535681153754756