| L(s) = 1 | + (−0.740 − 0.0554i)2-s + (0.578 + 1.87i)3-s + (−1.43 − 0.215i)4-s + (1.15 − 1.91i)5-s + (−0.324 − 1.42i)6-s + (2.23 − 1.41i)7-s + (2.49 + 0.569i)8-s + (−0.708 + 0.482i)9-s + (−0.962 + 1.35i)10-s + (−0.619 − 0.422i)11-s + (−0.424 − 2.81i)12-s + (1.37 + 2.85i)13-s + (−1.73 + 0.922i)14-s + (4.26 + 1.06i)15-s + (0.952 + 0.293i)16-s + (3.02 + 1.18i)17-s + ⋯ |
| L(s) = 1 | + (−0.523 − 0.0392i)2-s + (0.334 + 1.08i)3-s + (−0.716 − 0.107i)4-s + (0.516 − 0.856i)5-s + (−0.132 − 0.580i)6-s + (0.845 − 0.534i)7-s + (0.882 + 0.201i)8-s + (−0.236 + 0.160i)9-s + (−0.304 + 0.427i)10-s + (−0.186 − 0.127i)11-s + (−0.122 − 0.812i)12-s + (0.381 + 0.792i)13-s + (−0.463 + 0.246i)14-s + (1.10 + 0.274i)15-s + (0.238 + 0.0734i)16-s + (0.734 + 0.288i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.956 - 0.291i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.956 - 0.291i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.08886 + 0.162394i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.08886 + 0.162394i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-1.15 + 1.91i)T \) |
| 7 | \( 1 + (-2.23 + 1.41i)T \) |
| good | 2 | \( 1 + (0.740 + 0.0554i)T + (1.97 + 0.298i)T^{2} \) |
| 3 | \( 1 + (-0.578 - 1.87i)T + (-2.47 + 1.68i)T^{2} \) |
| 11 | \( 1 + (0.619 + 0.422i)T + (4.01 + 10.2i)T^{2} \) |
| 13 | \( 1 + (-1.37 - 2.85i)T + (-8.10 + 10.1i)T^{2} \) |
| 17 | \( 1 + (-3.02 - 1.18i)T + (12.4 + 11.5i)T^{2} \) |
| 19 | \( 1 + (1.20 - 2.09i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.367 + 0.144i)T + (16.8 - 15.6i)T^{2} \) |
| 29 | \( 1 + (-0.616 - 0.772i)T + (-6.45 + 28.2i)T^{2} \) |
| 31 | \( 1 + (0.670 + 1.16i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (0.165 + 1.09i)T + (-35.3 + 10.9i)T^{2} \) |
| 41 | \( 1 + (-2.63 + 11.5i)T + (-36.9 - 17.7i)T^{2} \) |
| 43 | \( 1 + (5.54 - 1.26i)T + (38.7 - 18.6i)T^{2} \) |
| 47 | \( 1 + (-7.29 - 0.547i)T + (46.4 + 7.00i)T^{2} \) |
| 53 | \( 1 + (-0.187 + 1.24i)T + (-50.6 - 15.6i)T^{2} \) |
| 59 | \( 1 + (9.19 - 8.53i)T + (4.40 - 58.8i)T^{2} \) |
| 61 | \( 1 + (12.2 - 1.83i)T + (58.2 - 17.9i)T^{2} \) |
| 67 | \( 1 + (2.80 - 1.61i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (3.77 - 4.72i)T + (-15.7 - 69.2i)T^{2} \) |
| 73 | \( 1 + (13.3 - 1.00i)T + (72.1 - 10.8i)T^{2} \) |
| 79 | \( 1 + (6.82 - 11.8i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-7.77 + 16.1i)T + (-51.7 - 64.8i)T^{2} \) |
| 89 | \( 1 + (-5.66 + 3.86i)T + (32.5 - 82.8i)T^{2} \) |
| 97 | \( 1 - 2.64iT - 97T^{2} \) |
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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
−12.08626855764753707240807289595, −10.63702126333030356001080069444, −10.18456067254555724516919427694, −9.116106007173653541892813150692, −8.695269980925844700652745160574, −7.60241121273728373567777120775, −5.65665854505284581854270130901, −4.61462982803855452800979767886, −3.95488514942775828663782177465, −1.44146794069147985100063554503,
1.49866009784916632650482990228, 2.93961299136767107337966654818, 4.86798100110376170924235952363, 6.13392310622691965197630094101, 7.46270804272684025925560916730, 7.969953442179897153104892075536, 8.973518970997643966357166570748, 10.07927336771667872114568291248, 10.94785335739650198275892505644, 12.19512567923243631433897840054
