L(s) = 1 | + (−0.809 − 0.587i)11-s + (0.5 − 1.53i)17-s + (−1.30 − 0.951i)19-s + (−0.809 − 0.587i)25-s + (0.5 + 0.363i)41-s − 0.618·43-s + (0.309 − 0.951i)49-s + (1.30 − 0.951i)59-s + 1.61·67-s + (−0.5 + 0.363i)73-s + (0.190 − 0.587i)83-s − 0.618·89-s + (0.190 + 0.587i)97-s + (−0.5 − 0.363i)107-s + (−1.30 − 0.951i)113-s + ⋯ |
L(s) = 1 | + (−0.809 − 0.587i)11-s + (0.5 − 1.53i)17-s + (−1.30 − 0.951i)19-s + (−0.809 − 0.587i)25-s + (0.5 + 0.363i)41-s − 0.618·43-s + (0.309 − 0.951i)49-s + (1.30 − 0.951i)59-s + 1.61·67-s + (−0.5 + 0.363i)73-s + (0.190 − 0.587i)83-s − 0.618·89-s + (0.190 + 0.587i)97-s + (−0.5 − 0.363i)107-s + (−1.30 − 0.951i)113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0457 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0457 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9016144819\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9016144819\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 + (0.809 + 0.587i)T \) |
good | 5 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 7 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 13 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 17 | \( 1 + (-0.5 + 1.53i)T + (-0.809 - 0.587i)T^{2} \) |
| 19 | \( 1 + (1.30 + 0.951i)T + (0.309 + 0.951i)T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 31 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 37 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 41 | \( 1 + (-0.5 - 0.363i)T + (0.309 + 0.951i)T^{2} \) |
| 43 | \( 1 + 0.618T + T^{2} \) |
| 47 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 53 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 59 | \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \) |
| 61 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 67 | \( 1 - 1.61T + T^{2} \) |
| 71 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 73 | \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \) |
| 79 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 83 | \( 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2} \) |
| 89 | \( 1 + 0.618T + T^{2} \) |
| 97 | \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \) |
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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
−8.543756499614853776384927967640, −8.034152575219089111627334674913, −7.13885633217300881483971103300, −6.48576770165856396331977212650, −5.51730061445696420466940136793, −4.91940656747496446365433150814, −3.97053611754875168791648848939, −2.91659900908040223152624716826, −2.20947650309939078163285985354, −0.52774960880430843057271949369,
1.59146278825758819847380335067, 2.43153751801673521800432644056, 3.68891216133920004429255567296, 4.25391550577294207248315903876, 5.37108736125658574604079666869, 5.95823583826691259322740043329, 6.78117654417052377355981495150, 7.75925917163788792281159894185, 8.157700318260062138742506583037, 8.967505721880164930608161700082