L(s) = 1 | + (0.374 + 0.927i)3-s + (−0.882 + 0.469i)4-s + (−0.542 − 1.89i)5-s + (−0.719 + 0.694i)9-s + (−0.766 − 0.642i)12-s + (1.55 − 1.21i)15-s + (0.559 − 0.829i)16-s + (1.36 + 1.41i)20-s + (−1.70 + 0.300i)23-s + (−2.44 + 1.52i)25-s + (−0.913 − 0.406i)27-s + (−0.671 + 1.37i)31-s + (0.309 − 0.951i)36-s + (−0.232 + 0.258i)37-s + (1.70 + 0.984i)45-s + ⋯ |
L(s) = 1 | + (0.374 + 0.927i)3-s + (−0.882 + 0.469i)4-s + (−0.542 − 1.89i)5-s + (−0.719 + 0.694i)9-s + (−0.766 − 0.642i)12-s + (1.55 − 1.21i)15-s + (0.559 − 0.829i)16-s + (1.36 + 1.41i)20-s + (−1.70 + 0.300i)23-s + (−2.44 + 1.52i)25-s + (−0.913 − 0.406i)27-s + (−0.671 + 1.37i)31-s + (0.309 − 0.951i)36-s + (−0.232 + 0.258i)37-s + (1.70 + 0.984i)45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 + 0.150i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 + 0.150i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.04519643734\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.04519643734\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.374 - 0.927i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (0.882 - 0.469i)T^{2} \) |
| 5 | \( 1 + (0.542 + 1.89i)T + (-0.848 + 0.529i)T^{2} \) |
| 7 | \( 1 + (0.961 - 0.275i)T^{2} \) |
| 13 | \( 1 + (0.990 - 0.139i)T^{2} \) |
| 17 | \( 1 + (-0.669 - 0.743i)T^{2} \) |
| 19 | \( 1 + (-0.104 + 0.994i)T^{2} \) |
| 23 | \( 1 + (1.70 - 0.300i)T + (0.939 - 0.342i)T^{2} \) |
| 29 | \( 1 + (0.719 + 0.694i)T^{2} \) |
| 31 | \( 1 + (0.671 - 1.37i)T + (-0.615 - 0.788i)T^{2} \) |
| 37 | \( 1 + (0.232 - 0.258i)T + (-0.104 - 0.994i)T^{2} \) |
| 41 | \( 1 + (0.719 - 0.694i)T^{2} \) |
| 43 | \( 1 + (0.766 - 0.642i)T^{2} \) |
| 47 | \( 1 + (-0.321 + 0.603i)T + (-0.559 - 0.829i)T^{2} \) |
| 53 | \( 1 + (-0.755 + 1.04i)T + (-0.309 - 0.951i)T^{2} \) |
| 59 | \( 1 + (1.28 + 0.0448i)T + (0.997 + 0.0697i)T^{2} \) |
| 61 | \( 1 + (-0.615 + 0.788i)T^{2} \) |
| 67 | \( 1 + (1.17 - 0.984i)T + (0.173 - 0.984i)T^{2} \) |
| 71 | \( 1 + (0.278 - 0.624i)T + (-0.669 - 0.743i)T^{2} \) |
| 73 | \( 1 + (0.913 + 0.406i)T^{2} \) |
| 79 | \( 1 + (-0.882 + 0.469i)T^{2} \) |
| 83 | \( 1 + (-0.990 - 0.139i)T^{2} \) |
| 89 | \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + (1.80 + 0.518i)T + (0.848 + 0.529i)T^{2} \) |
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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
−9.103343717788894458027414638039, −8.450526665275352277249714900577, −8.213695814766061714417090871970, −7.37431797546830768770370943500, −5.70525730314851929544015929737, −5.21459587897563653451881810478, −4.43700038984622771974940066194, −4.03383807456461692156366025750, −3.22556132019212632215869645704, −1.62508953193746238429495315226,
0.02576460796519395051853795457, 1.84010405668305246494044459179, 2.74253671099325749754971100913, 3.64203059581268271444200054209, 4.27819190191630532555512585522, 5.83574502119993258235836196288, 6.15125789695300628384668773094, 7.00739933978817343240116885277, 7.77264892136879581961604283146, 8.120647554799933037995466130544