L(s) = 1 | + (−0.5 + 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.499 − 0.866i)4-s + (0.499 + 0.866i)6-s + 0.999·8-s − 11-s − 0.999·12-s + (−0.5 + 0.866i)16-s + (−1 + 1.73i)17-s + (−0.5 − 0.866i)19-s + (0.5 − 0.866i)22-s + (0.5 − 0.866i)24-s + (−0.5 − 0.866i)25-s + 27-s + (−0.499 − 0.866i)32-s + (−0.5 + 0.866i)33-s + ⋯ |
L(s) = 1 | + (−0.5 + 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.499 − 0.866i)4-s + (0.499 + 0.866i)6-s + 0.999·8-s − 11-s − 0.999·12-s + (−0.5 + 0.866i)16-s + (−1 + 1.73i)17-s + (−0.5 − 0.866i)19-s + (0.5 − 0.866i)22-s + (0.5 − 0.866i)24-s + (−0.5 − 0.866i)25-s + 27-s + (−0.499 − 0.866i)32-s + (−0.5 + 0.866i)33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 - 0.211i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 - 0.211i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5546383861\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5546383861\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 19 | \( 1 + (0.5 + 0.866i)T \) |
good | 3 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 5 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 + T + T^{2} \) |
| 13 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 73 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + T + T^{2} \) |
| 89 | \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 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
−13.25814109219072467176254385597, −12.81533261395288879606324937369, −10.98704515428475493520638789862, −10.12876269430643243027513528919, −8.657723995035487315844952114077, −8.101073976834170344302537445441, −7.05485255925857981371645894753, −6.06133272168678729103772022551, −4.54406087258176959610073463060, −2.12233152739616025475607167415,
2.56999500303935729279473701049, 3.85397393163801438416186496316, 5.05189013214110699027880545501, 7.18947349529932496515057948897, 8.410114434551979520735328937624, 9.327406285338906020916266976238, 10.08373028466783430772051515063, 10.98317146030726192002502408597, 12.02361452398343865595862925576, 13.15459683331786685532889254614