L(s) = 1 | + (0.241 − 0.970i)3-s + (0.438 − 0.898i)4-s + (−1.96 − 0.0687i)5-s + (−0.882 − 0.469i)9-s + (−0.766 − 0.642i)12-s + (−0.542 + 1.89i)15-s + (−0.615 − 0.788i)16-s + (−0.924 + 1.73i)20-s + (−1.70 + 0.300i)23-s + (2.87 + 0.200i)25-s + (−0.669 + 0.743i)27-s + (−1.51 − 0.213i)31-s + (−0.809 + 0.587i)36-s + (0.0363 − 0.345i)37-s + (1.70 + 0.984i)45-s + ⋯ |
L(s) = 1 | + (0.241 − 0.970i)3-s + (0.438 − 0.898i)4-s + (−1.96 − 0.0687i)5-s + (−0.882 − 0.469i)9-s + (−0.766 − 0.642i)12-s + (−0.542 + 1.89i)15-s + (−0.615 − 0.788i)16-s + (−0.924 + 1.73i)20-s + (−1.70 + 0.300i)23-s + (2.87 + 0.200i)25-s + (−0.669 + 0.743i)27-s + (−1.51 − 0.213i)31-s + (−0.809 + 0.587i)36-s + (0.0363 − 0.345i)37-s + (1.70 + 0.984i)45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.491 - 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.491 - 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2758085399\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2758085399\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.241 + 0.970i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-0.438 + 0.898i)T^{2} \) |
| 5 | \( 1 + (1.96 + 0.0687i)T + (0.997 + 0.0697i)T^{2} \) |
| 7 | \( 1 + (0.0348 - 0.999i)T^{2} \) |
| 13 | \( 1 + (-0.719 + 0.694i)T^{2} \) |
| 17 | \( 1 + (0.104 + 0.994i)T^{2} \) |
| 19 | \( 1 + (-0.978 + 0.207i)T^{2} \) |
| 23 | \( 1 + (1.70 - 0.300i)T + (0.939 - 0.342i)T^{2} \) |
| 29 | \( 1 + (0.882 - 0.469i)T^{2} \) |
| 31 | \( 1 + (1.51 + 0.213i)T + (0.961 + 0.275i)T^{2} \) |
| 37 | \( 1 + (-0.0363 + 0.345i)T + (-0.978 - 0.207i)T^{2} \) |
| 41 | \( 1 + (0.882 + 0.469i)T^{2} \) |
| 43 | \( 1 + (0.766 - 0.642i)T^{2} \) |
| 47 | \( 1 + (0.614 - 0.299i)T + (0.615 - 0.788i)T^{2} \) |
| 53 | \( 1 + (-1.22 - 0.397i)T + (0.809 + 0.587i)T^{2} \) |
| 59 | \( 1 + (-1.06 + 0.718i)T + (0.374 - 0.927i)T^{2} \) |
| 61 | \( 1 + (0.961 - 0.275i)T^{2} \) |
| 67 | \( 1 + (1.17 - 0.984i)T + (0.173 - 0.984i)T^{2} \) |
| 71 | \( 1 + (-0.508 - 0.457i)T + (0.104 + 0.994i)T^{2} \) |
| 73 | \( 1 + (0.669 - 0.743i)T^{2} \) |
| 79 | \( 1 + (0.438 - 0.898i)T^{2} \) |
| 83 | \( 1 + (0.719 + 0.694i)T^{2} \) |
| 89 | \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + (0.0655 + 1.87i)T + (-0.997 + 0.0697i)T^{2} \) |
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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
−8.197906420535319666694901939006, −7.39590029307197950919730838659, −7.16618536868887022360716075060, −6.16770479339924260770515586204, −5.43820687010410431390736960341, −4.31040202153039419753588723407, −3.56628863914907215023739515600, −2.53793004300170031688407726671, −1.41547899508825212609182599351, −0.15806215625776745071411809700,
2.31684592081331610496428044213, 3.35383730531335035300633659935, 3.81162484224979743829430265034, 4.32547659975166832220864256477, 5.29589344382228591030994932870, 6.54019061207758440162275531246, 7.35394719184436487382024757182, 7.958484685278723874152546622203, 8.427293423619262093139259463748, 9.032827693254068163363318089239