| L(s) = 1 | + (0.0743 + 1.98i)2-s + (−0.153 − 0.207i)3-s + (−1.95 + 0.146i)4-s + (1.96 + 1.07i)5-s + (0.401 − 0.319i)6-s + (2.51 + 0.823i)7-s + (0.00934 + 0.0829i)8-s + (0.864 − 2.80i)9-s + (−1.98 + 3.98i)10-s + (−2.75 + 0.849i)11-s + (0.329 + 0.382i)12-s + (3.20 − 2.01i)13-s + (−1.45 + 5.06i)14-s + (−0.0783 − 0.571i)15-s + (−4.03 + 0.608i)16-s + (−6.03 + 1.14i)17-s + ⋯ |
| L(s) = 1 | + (0.0526 + 1.40i)2-s + (−0.0883 − 0.119i)3-s + (−0.976 + 0.0731i)4-s + (0.877 + 0.479i)5-s + (0.163 − 0.130i)6-s + (0.950 + 0.311i)7-s + (0.00330 + 0.0293i)8-s + (0.288 − 0.934i)9-s + (−0.627 + 1.25i)10-s + (−0.830 + 0.256i)11-s + (0.0950 + 0.110i)12-s + (0.888 − 0.558i)13-s + (−0.387 + 1.35i)14-s + (−0.0202 − 0.147i)15-s + (−1.00 + 0.152i)16-s + (−1.46 + 0.276i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.412 - 0.910i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.412 - 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.804137 + 1.24731i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.804137 + 1.24731i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-1.96 - 1.07i)T \) |
| 7 | \( 1 + (-2.51 - 0.823i)T \) |
| good | 2 | \( 1 + (-0.0743 - 1.98i)T + (-1.99 + 0.149i)T^{2} \) |
| 3 | \( 1 + (0.153 + 0.207i)T + (-0.884 + 2.86i)T^{2} \) |
| 11 | \( 1 + (2.75 - 0.849i)T + (9.08 - 6.19i)T^{2} \) |
| 13 | \( 1 + (-3.20 + 2.01i)T + (5.64 - 11.7i)T^{2} \) |
| 17 | \( 1 + (6.03 - 1.14i)T + (15.8 - 6.21i)T^{2} \) |
| 19 | \( 1 + (1.91 - 3.31i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.465 + 2.46i)T + (-21.4 - 8.40i)T^{2} \) |
| 29 | \( 1 + (1.94 + 4.03i)T + (-18.0 + 22.6i)T^{2} \) |
| 31 | \( 1 + (-2.00 + 1.15i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-8.62 + 7.42i)T + (5.51 - 36.5i)T^{2} \) |
| 41 | \( 1 + (6.23 + 4.97i)T + (9.12 + 39.9i)T^{2} \) |
| 43 | \( 1 + (-3.59 - 0.404i)T + (41.9 + 9.56i)T^{2} \) |
| 47 | \( 1 + (-0.971 + 0.0363i)T + (46.8 - 3.51i)T^{2} \) |
| 53 | \( 1 + (-0.0807 - 0.0695i)T + (7.89 + 52.4i)T^{2} \) |
| 59 | \( 1 + (4.29 - 10.9i)T + (-43.2 - 40.1i)T^{2} \) |
| 61 | \( 1 + (2.86 + 0.214i)T + (60.3 + 9.09i)T^{2} \) |
| 67 | \( 1 + (-2.08 + 7.76i)T + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + (-12.4 - 5.99i)T + (44.2 + 55.5i)T^{2} \) |
| 73 | \( 1 + (4.74 + 0.177i)T + (72.7 + 5.45i)T^{2} \) |
| 79 | \( 1 + (9.73 + 5.61i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-3.18 + 5.07i)T + (-36.0 - 74.7i)T^{2} \) |
| 89 | \( 1 + (6.32 + 1.94i)T + (73.5 + 50.1i)T^{2} \) |
| 97 | \( 1 + (1.33 - 1.33i)T - 97iT^{2} \) |
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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
−12.74299688647712062004024606984, −11.28843152177272850848613391583, −10.49254775461485473001921533303, −9.139487262879906300316640851147, −8.307724786989596659419055576889, −7.30181081989452719207712353291, −6.21778008079781747777982477206, −5.71271415108975394949925805086, −4.36494739714898197105691531971, −2.21260853162121834312457413041,
1.50022295282407853412439434572, 2.52522683983296834650573388049, 4.38102566880154153640971132277, 5.06325621768173724392973673948, 6.68831014968720002341815271654, 8.221608834945810488750418097343, 9.151107528029556430724960149948, 10.19828044674991747307531008296, 11.06456266800513645038433993906, 11.32640828330918121872799531984
