| L(s) = 1 | + (−2.10 + 0.937i)2-s + (−0.669 − 0.743i)3-s + (2.21 − 2.46i)4-s + (1.88 − 1.36i)5-s + (2.10 + 0.937i)6-s + (−1.97 + 2.19i)7-s + (−0.938 + 2.88i)8-s + (−0.104 + 0.994i)9-s + (−2.68 + 4.65i)10-s + (3.31 + 0.0356i)11-s − 3.31·12-s + (−1.80 + 3.12i)13-s + (2.10 − 6.48i)14-s + (−2.27 − 0.484i)15-s + (−0.0388 − 0.369i)16-s + (−4.40 − 1.96i)17-s + ⋯ |
| L(s) = 1 | + (−1.48 + 0.663i)2-s + (−0.386 − 0.429i)3-s + (1.10 − 1.23i)4-s + (0.842 − 0.612i)5-s + (0.860 + 0.382i)6-s + (−0.747 + 0.830i)7-s + (−0.331 + 1.02i)8-s + (−0.0348 + 0.331i)9-s + (−0.849 + 1.47i)10-s + (0.999 + 0.0107i)11-s − 0.957·12-s + (−0.500 + 0.865i)13-s + (0.563 − 1.73i)14-s + (−0.588 − 0.125i)15-s + (−0.00971 − 0.0923i)16-s + (−1.06 − 0.475i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.931 - 0.363i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.931 - 0.363i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.633640 + 0.119411i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.633640 + 0.119411i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.669 + 0.743i)T \) |
| 11 | \( 1 + (-3.31 - 0.0356i)T \) |
| 13 | \( 1 + (1.80 - 3.12i)T \) |
| good | 2 | \( 1 + (2.10 - 0.937i)T + (1.33 - 1.48i)T^{2} \) |
| 5 | \( 1 + (-1.88 + 1.36i)T + (1.54 - 4.75i)T^{2} \) |
| 7 | \( 1 + (1.97 - 2.19i)T + (-0.731 - 6.96i)T^{2} \) |
| 17 | \( 1 + (4.40 + 1.96i)T + (11.3 + 12.6i)T^{2} \) |
| 19 | \( 1 + (-6.89 + 1.46i)T + (17.3 - 7.72i)T^{2} \) |
| 23 | \( 1 + (-2.81 + 4.88i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-7.02 - 1.49i)T + (26.4 + 11.7i)T^{2} \) |
| 31 | \( 1 + (4.32 + 3.14i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (-8.14 - 1.73i)T + (33.8 + 15.0i)T^{2} \) |
| 41 | \( 1 + (-2.99 - 3.32i)T + (-4.28 + 40.7i)T^{2} \) |
| 43 | \( 1 + (-4.41 - 7.64i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-1.23 + 3.80i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (-7.37 - 5.35i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (-1.62 + 1.80i)T + (-6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 + (2.51 + 1.12i)T + (40.8 + 45.3i)T^{2} \) |
| 67 | \( 1 + (-6.63 + 11.4i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-2.79 - 1.24i)T + (47.5 + 52.7i)T^{2} \) |
| 73 | \( 1 + (2.74 + 8.45i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (5.21 + 3.79i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (-13.1 + 9.54i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (3.65 - 6.33i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (0.363 - 3.45i)T + (-94.8 - 20.1i)T^{2} \) |
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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
−11.08534875785014512121776061369, −9.729703421127318500305844368680, −9.275150169308437301531169899272, −8.857430873652696671167803614842, −7.48972474617759126080850118445, −6.57371737070417765422836741048, −6.08725378993667583986201891608, −4.80322106798493908850131166843, −2.41101945712865008765108541650, −1.02072258856009683193393923902,
0.975961987174675119950214279248, 2.63186358284784571333966391746, 3.78171883791259821798492459364, 5.57054295910442693023572375719, 6.75066184942332665362325986183, 7.43157144654288801204234278912, 8.810240230799543492432040149435, 9.690636307559068409026717350742, 10.02578450492994253426377563225, 10.78694388520312367932589148127
