L(s) = 1 | + (−0.0216 − 1.73i)3-s + (1.06 + 1.96i)5-s + (0.0973 − 2.64i)7-s + (−2.99 + 0.0749i)9-s + (2.62 − 1.51i)11-s + 2.31·13-s + (3.38 − 1.89i)15-s + (4.49 − 2.59i)17-s + (−5.58 − 3.22i)19-s + (−4.58 − 0.111i)21-s + (2.43 − 4.21i)23-s + (−2.72 + 4.19i)25-s + (0.194 + 5.19i)27-s + 3.48i·29-s + (1.16 − 0.673i)31-s + ⋯ |
L(s) = 1 | + (−0.0124 − 0.999i)3-s + (0.477 + 0.878i)5-s + (0.0368 − 0.999i)7-s + (−0.999 + 0.0249i)9-s + (0.792 − 0.457i)11-s + 0.642·13-s + (0.872 − 0.488i)15-s + (1.09 − 0.629i)17-s + (−1.28 − 0.739i)19-s + (−0.999 − 0.0243i)21-s + (0.507 − 0.879i)23-s + (−0.544 + 0.838i)25-s + (0.0374 + 0.999i)27-s + 0.647i·29-s + (0.209 − 0.120i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.339 + 0.940i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.339 + 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.22102 - 0.857409i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.22102 - 0.857409i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.0216 + 1.73i)T \) |
| 5 | \( 1 + (-1.06 - 1.96i)T \) |
| 7 | \( 1 + (-0.0973 + 2.64i)T \) |
good | 11 | \( 1 + (-2.62 + 1.51i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 2.31T + 13T^{2} \) |
| 17 | \( 1 + (-4.49 + 2.59i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (5.58 + 3.22i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.43 + 4.21i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 3.48iT - 29T^{2} \) |
| 31 | \( 1 + (-1.16 + 0.673i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (2.43 + 1.40i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 1.06T + 41T^{2} \) |
| 43 | \( 1 + 2.42iT - 43T^{2} \) |
| 47 | \( 1 + (-3.30 - 1.90i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (3.52 + 6.11i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-6.18 - 10.7i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-11.4 - 6.60i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (3.72 - 2.14i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 7.08iT - 71T^{2} \) |
| 73 | \( 1 + (0.0763 + 0.132i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (3.04 - 5.27i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 10.2iT - 83T^{2} \) |
| 89 | \( 1 + (7.10 - 12.3i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 8.63T + 97T^{2} \) |
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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
−11.00523698591492381886626449434, −10.37548897258659360000316555127, −9.101112414590612641663859930908, −8.156511602512564406447122174445, −6.93574084555657698118913778029, −6.69519261056291404600729533168, −5.52720898998153866716964314878, −3.83690598324089836234090358641, −2.65352368957851294277261617047, −1.08873839232801993477037541209,
1.80604702171426812395808950188, 3.50987099789945658601533252548, 4.57471296952349065128665578428, 5.62151317588868216960469124066, 6.23261599431086934483769320700, 8.146706179535067569558275398292, 8.769049496722226029405419700657, 9.550695552531550596482363497131, 10.24212260089879968552550177620, 11.41390659116737923382608920261