L(s) = 1 | + (−0.955 + 0.294i)2-s + (0.826 − 0.563i)4-s + (−0.826 − 0.563i)7-s + (−0.623 + 0.781i)8-s + (−0.733 + 0.680i)9-s + (0.535 + 0.496i)11-s + (−0.222 + 0.974i)13-s + (0.955 + 0.294i)14-s + (0.365 − 0.930i)16-s + (0.142 − 1.90i)17-s + (0.5 − 0.866i)18-s + (−0.733 − 1.26i)19-s + (−0.658 − 0.317i)22-s + (0.955 + 0.294i)25-s + (−0.0747 − 0.997i)26-s + ⋯ |
L(s) = 1 | + (−0.955 + 0.294i)2-s + (0.826 − 0.563i)4-s + (−0.826 − 0.563i)7-s + (−0.623 + 0.781i)8-s + (−0.733 + 0.680i)9-s + (0.535 + 0.496i)11-s + (−0.222 + 0.974i)13-s + (0.955 + 0.294i)14-s + (0.365 − 0.930i)16-s + (0.142 − 1.90i)17-s + (0.5 − 0.866i)18-s + (−0.733 − 1.26i)19-s + (−0.658 − 0.317i)22-s + (0.955 + 0.294i)25-s + (−0.0747 − 0.997i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.942 + 0.335i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.942 + 0.335i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6398259093\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6398259093\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.955 - 0.294i)T \) |
| 7 | \( 1 + (0.826 + 0.563i)T \) |
| 13 | \( 1 + (0.222 - 0.974i)T \) |
good | 3 | \( 1 + (0.733 - 0.680i)T^{2} \) |
| 5 | \( 1 + (-0.955 - 0.294i)T^{2} \) |
| 11 | \( 1 + (-0.535 - 0.496i)T + (0.0747 + 0.997i)T^{2} \) |
| 17 | \( 1 + (-0.142 + 1.90i)T + (-0.988 - 0.149i)T^{2} \) |
| 19 | \( 1 + (0.733 + 1.26i)T + (-0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (0.988 - 0.149i)T^{2} \) |
| 29 | \( 1 + (-1.78 + 0.858i)T + (0.623 - 0.781i)T^{2} \) |
| 31 | \( 1 + (0.222 - 0.385i)T + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.365 - 0.930i)T^{2} \) |
| 41 | \( 1 + (0.222 + 0.974i)T^{2} \) |
| 43 | \( 1 + (0.222 - 0.974i)T^{2} \) |
| 47 | \( 1 + (-1.88 + 0.582i)T + (0.826 - 0.563i)T^{2} \) |
| 53 | \( 1 + (0.826 - 0.563i)T + (0.365 - 0.930i)T^{2} \) |
| 59 | \( 1 + (-1.63 + 0.246i)T + (0.955 - 0.294i)T^{2} \) |
| 61 | \( 1 + (0.826 + 0.563i)T + (0.365 + 0.930i)T^{2} \) |
| 67 | \( 1 + (-0.0747 + 0.129i)T + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (-1.72 - 0.829i)T + (0.623 + 0.781i)T^{2} \) |
| 73 | \( 1 + (-0.826 - 0.563i)T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (0.400 + 1.75i)T + (-0.900 + 0.433i)T^{2} \) |
| 89 | \( 1 + (-0.0747 + 0.997i)T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.130719061672923043685370450122, −8.436652379702907805273509535420, −7.32335894474094311180186224056, −6.93695773489676212480276789512, −6.35069250027195575477413305445, −5.15336896968513287155707911905, −4.46079976208898337606423269269, −2.93300900298812348688110475679, −2.32760106054431835396629145538, −0.71031078871356481121433447488,
1.02954673507593643672465603530, 2.43129427400672876254454075351, 3.28721983630939741140046362539, 3.90030967976046433439745280913, 5.63798784780453617774746441420, 6.21961740792683510724527580187, 6.68834055631876256465597611621, 7.985121330669656893288118695583, 8.540829906690201730051119147615, 8.919125958457076623972746596701