L(s) = 1 | + (0.5 + 0.866i)5-s + (−0.5 + 0.866i)7-s + (−0.5 + 0.866i)11-s + (−1 + 1.73i)29-s + (−0.5 − 0.866i)31-s − 0.999·35-s − 53-s − 0.999·55-s + (1 + 1.73i)59-s − 73-s + (−0.499 − 0.866i)77-s + (1 − 1.73i)79-s + (−0.5 + 0.866i)83-s + (0.5 − 0.866i)97-s + (0.5 − 0.866i)101-s + ⋯ |
L(s) = 1 | + (0.5 + 0.866i)5-s + (−0.5 + 0.866i)7-s + (−0.5 + 0.866i)11-s + (−1 + 1.73i)29-s + (−0.5 − 0.866i)31-s − 0.999·35-s − 53-s − 0.999·55-s + (1 + 1.73i)59-s − 73-s + (−0.499 − 0.866i)77-s + (1 − 1.73i)79-s + (−0.5 + 0.866i)83-s + (0.5 − 0.866i)97-s + (0.5 − 0.866i)101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.033274016\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.033274016\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 7 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + T + T^{2} \) |
| 59 | \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + T + T^{2} \) |
| 79 | \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 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
−9.308421185835438886048758043886, −8.720451435882704175613158966602, −7.56761468696955505465284696320, −7.05521667031257861121735904743, −6.16677892930053194207332949836, −5.58076918878965521926601969280, −4.66280258585824677256178483758, −3.45246061285854891360932999537, −2.64368893532343021020690490710, −1.87892607903539875032199093502,
0.64995692051352294619609908117, 1.92278553653261447378687378450, 3.19846511782465732387835815939, 4.00139006917924059981251685928, 4.99308920425133232554012307526, 5.68832144500077914025905953717, 6.45859589274929203428130007560, 7.37340318901150542981047551625, 8.123739647569303933861357751369, 8.856196251765182350681314374375