L(s) = 1 | + (0.5 + 0.866i)3-s + (−0.499 + 0.866i)9-s + (−0.5 − 0.866i)11-s − 17-s + 19-s + (−0.5 − 0.866i)25-s − 0.999·27-s + (0.499 − 0.866i)33-s + (0.5 − 0.866i)41-s + (−0.5 − 0.866i)43-s + (−0.5 + 0.866i)49-s + (−0.5 − 0.866i)51-s + (0.5 + 0.866i)57-s + (−0.5 + 0.866i)59-s + (−0.5 + 0.866i)67-s + ⋯ |
L(s) = 1 | + (0.5 + 0.866i)3-s + (−0.499 + 0.866i)9-s + (−0.5 − 0.866i)11-s − 17-s + 19-s + (−0.5 − 0.866i)25-s − 0.999·27-s + (0.499 − 0.866i)33-s + (0.5 − 0.866i)41-s + (−0.5 − 0.866i)43-s + (−0.5 + 0.866i)49-s + (−0.5 − 0.866i)51-s + (0.5 + 0.866i)57-s + (−0.5 + 0.866i)59-s + (−0.5 + 0.866i)67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8491845197\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8491845197\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.5 - 0.866i)T \) |
good | 5 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 7 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 + T + T^{2} \) |
| 19 | \( 1 - T + T^{2} \) |
| 23 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 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 | \( 1 - T^{2} \) |
| 73 | \( 1 + T + T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 - 2T + T^{2} \) |
| 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
−11.97573815334376466413570916088, −11.01712646271616680217059782019, −10.30123294405171983945319577700, −9.265452115718542584533803237753, −8.492488601413409686406975514725, −7.52059976324434673095561067634, −6.02084406712930474359488698716, −4.94752894790484327388806822254, −3.76292847751230084723448605318, −2.55740197511479142388327182984,
1.88464288874454619446146495074, 3.23161892249714255438115202390, 4.80648808073003878039043101676, 6.17276682240831828685795082405, 7.22493652639742857624836174268, 7.917735007051806506900779299386, 9.063727139643326893947662893369, 9.868532226469307474670523599663, 11.20665783989842032015504872554, 12.01611523951067512914040256204