L(s) = 1 | + (−0.365 − 0.930i)7-s + (−1.55 − 1.24i)13-s + (−1.61 − 0.930i)19-s + (0.365 − 0.930i)25-s + (1.17 − 0.680i)31-s + (1.40 + 1.29i)37-s + (−1.78 + 0.858i)43-s + (−0.733 + 0.680i)49-s + (1.32 − 1.42i)61-s + (−0.826 − 1.43i)67-s + (0.548 + 0.215i)73-s + (0.733 − 1.26i)79-s + (−0.587 + 1.90i)91-s + 1.56i·97-s + (−0.167 − 0.246i)103-s + ⋯ |
L(s) = 1 | + (−0.365 − 0.930i)7-s + (−1.55 − 1.24i)13-s + (−1.61 − 0.930i)19-s + (0.365 − 0.930i)25-s + (1.17 − 0.680i)31-s + (1.40 + 1.29i)37-s + (−1.78 + 0.858i)43-s + (−0.733 + 0.680i)49-s + (1.32 − 1.42i)61-s + (−0.826 − 1.43i)67-s + (0.548 + 0.215i)73-s + (0.733 − 1.26i)79-s + (−0.587 + 1.90i)91-s + 1.56i·97-s + (−0.167 − 0.246i)103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.274 + 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.274 + 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7653237019\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7653237019\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (0.365 + 0.930i)T \) |
good | 5 | \( 1 + (-0.365 + 0.930i)T^{2} \) |
| 11 | \( 1 + (0.955 + 0.294i)T^{2} \) |
| 13 | \( 1 + (1.55 + 1.24i)T + (0.222 + 0.974i)T^{2} \) |
| 17 | \( 1 + (-0.826 + 0.563i)T^{2} \) |
| 19 | \( 1 + (1.61 + 0.930i)T + (0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (0.826 + 0.563i)T^{2} \) |
| 29 | \( 1 + (-0.900 - 0.433i)T^{2} \) |
| 31 | \( 1 + (-1.17 + 0.680i)T + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (-1.40 - 1.29i)T + (0.0747 + 0.997i)T^{2} \) |
| 41 | \( 1 + (-0.623 - 0.781i)T^{2} \) |
| 43 | \( 1 + (1.78 - 0.858i)T + (0.623 - 0.781i)T^{2} \) |
| 47 | \( 1 + (0.733 - 0.680i)T^{2} \) |
| 53 | \( 1 + (0.0747 - 0.997i)T^{2} \) |
| 59 | \( 1 + (-0.365 - 0.930i)T^{2} \) |
| 61 | \( 1 + (-1.32 + 1.42i)T + (-0.0747 - 0.997i)T^{2} \) |
| 67 | \( 1 + (0.826 + 1.43i)T + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + (-0.900 + 0.433i)T^{2} \) |
| 73 | \( 1 + (-0.548 - 0.215i)T + (0.733 + 0.680i)T^{2} \) |
| 79 | \( 1 + (-0.733 + 1.26i)T + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (0.222 - 0.974i)T^{2} \) |
| 89 | \( 1 + (-0.955 + 0.294i)T^{2} \) |
| 97 | \( 1 - 1.56iT - T^{2} \) |
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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
−9.461734841124064988551166785497, −8.206417038410405124296802332101, −7.84914540446961035808838310766, −6.73719411044172293462028141027, −6.31599121857812702925562037982, −4.88491363054085725762209691712, −4.49825228991454747589270588122, −3.18421111788138313269918125658, −2.36801178250083128416213148718, −0.54408882113953902598296067263,
1.92719683531556375317729741188, 2.66999265226807694274770003678, 3.95875732969036900714205734194, 4.81488161385498421540891329398, 5.70969505752996765975276929971, 6.58917569887027356563718296683, 7.21204769002994785989324705132, 8.326852409993621433550167241234, 8.913597174780715938940791415255, 9.714490521548968212166206024888