L(s) = 1 | + 3-s + (0.5 + 0.866i)4-s + 9-s + (0.5 + 0.866i)12-s + (−0.5 − 0.866i)13-s + (−0.499 + 0.866i)16-s + 1.73i·17-s + (−1.5 + 0.866i)23-s + 27-s + (1.5 + 0.866i)29-s + (0.5 + 0.866i)36-s + (−0.5 − 0.866i)39-s + (1 − 1.73i)43-s + (−0.499 + 0.866i)48-s + (−0.5 − 0.866i)49-s + ⋯ |
L(s) = 1 | + 3-s + (0.5 + 0.866i)4-s + 9-s + (0.5 + 0.866i)12-s + (−0.5 − 0.866i)13-s + (−0.499 + 0.866i)16-s + 1.73i·17-s + (−1.5 + 0.866i)23-s + 27-s + (1.5 + 0.866i)29-s + (0.5 + 0.866i)36-s + (−0.5 − 0.866i)39-s + (1 − 1.73i)43-s + (−0.499 + 0.866i)48-s + (−0.5 − 0.866i)49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.951858078\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.951858078\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + (0.5 + 0.866i)T \) |
good | 2 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 7 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 - 1.73iT - T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + 1.73iT - T^{2} \) |
| 59 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + (0.5 + 0.866i)T^{2} \) |
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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
−8.652413244334396453544107483757, −8.312374516755770733425426926118, −7.73503466082006036443695530623, −6.96838665869194703548953374107, −6.20777470872312083621191869065, −5.10014227738254434226044130620, −3.87156994043140486267481632491, −3.56335907794005351837963543553, −2.49831245392009793428463772958, −1.74083266992747888707149059598,
1.16407494743949634103978667390, 2.43476575950368154038989713057, 2.74032081412975021635178061459, 4.32192066677180653193422690428, 4.68804668415531114189549644890, 5.94119926150063309653908041131, 6.64540413559197328850293174871, 7.35271946575011852512331160784, 8.012099382562616019296108178265, 9.011803129509337660087032524685