L(s) = 1 | + (0.5 + 0.866i)3-s + (−0.5 + 0.866i)4-s − 5-s + (−0.499 + 0.866i)9-s − 11-s − 0.999·12-s + (−0.5 − 0.866i)13-s + (−0.5 − 0.866i)15-s + (−0.499 − 0.866i)16-s + (−0.5 − 0.866i)17-s + (0.5 − 0.866i)20-s + 25-s − 0.999·27-s + (−1 + 1.73i)29-s + (−0.5 − 0.866i)33-s + ⋯ |
L(s) = 1 | + (0.5 + 0.866i)3-s + (−0.5 + 0.866i)4-s − 5-s + (−0.499 + 0.866i)9-s − 11-s − 0.999·12-s + (−0.5 − 0.866i)13-s + (−0.5 − 0.866i)15-s + (−0.499 − 0.866i)16-s + (−0.5 − 0.866i)17-s + (0.5 − 0.866i)20-s + 25-s − 0.999·27-s + (−1 + 1.73i)29-s + (−0.5 − 0.866i)33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.592 + 0.805i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.592 + 0.805i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1526202318\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1526202318\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.5 - 0.866i)T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + T + T^{2} \) |
| 13 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + T + T^{2} \) |
| 73 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
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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
−9.566671981594095622006622633838, −8.859568537909559257021859357092, −8.238585446417142323135468133399, −7.62313981504224925251360684452, −7.06772490830395878993606070776, −5.27048683863024817261348554411, −4.93516210497033310093924648800, −3.97150994849339384102716452276, −3.21809515910575157188508358647, −2.60676946862025879113476950774,
0.095927526189540525474787946843, 1.65548428536180489242955050933, 2.64682065646636589172963596635, 3.92035110448656786628469104350, 4.58087534062662674520746157327, 5.68998487486169094848447714222, 6.44522424421298641649971969984, 7.32035518829285188418376571392, 7.974274197515447138288387367757, 8.637373265685509589425506990638