L(s) = 1 | + (−0.156 − 0.987i)2-s + (0.987 + 0.156i)3-s + (−0.951 + 0.309i)4-s − i·6-s + (0.453 + 0.891i)8-s + (0.951 + 0.309i)9-s + (−0.119 − 0.101i)11-s + (−0.987 + 0.156i)12-s + (0.809 − 0.587i)16-s + (−0.144 − 1.84i)17-s + (0.156 − 0.987i)18-s + (0.891 + 0.546i)19-s + (−0.0819 + 0.133i)22-s + (0.309 + 0.951i)24-s + (0.587 + 0.809i)25-s + ⋯ |
L(s) = 1 | + (−0.156 − 0.987i)2-s + (0.987 + 0.156i)3-s + (−0.951 + 0.309i)4-s − i·6-s + (0.453 + 0.891i)8-s + (0.951 + 0.309i)9-s + (−0.119 − 0.101i)11-s + (−0.987 + 0.156i)12-s + (0.809 − 0.587i)16-s + (−0.144 − 1.84i)17-s + (0.156 − 0.987i)18-s + (0.891 + 0.546i)19-s + (−0.0819 + 0.133i)22-s + (0.309 + 0.951i)24-s + (0.587 + 0.809i)25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.508 + 0.861i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.508 + 0.861i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.213332789\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.213332789\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.156 + 0.987i)T \) |
| 3 | \( 1 + (-0.987 - 0.156i)T \) |
| 41 | \( 1 + (0.891 + 0.453i)T \) |
good | 5 | \( 1 + (-0.587 - 0.809i)T^{2} \) |
| 7 | \( 1 + (0.891 - 0.453i)T^{2} \) |
| 11 | \( 1 + (0.119 + 0.101i)T + (0.156 + 0.987i)T^{2} \) |
| 13 | \( 1 + (-0.453 + 0.891i)T^{2} \) |
| 17 | \( 1 + (0.144 + 1.84i)T + (-0.987 + 0.156i)T^{2} \) |
| 19 | \( 1 + (-0.891 - 0.546i)T + (0.453 + 0.891i)T^{2} \) |
| 23 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 29 | \( 1 + (0.987 + 0.156i)T^{2} \) |
| 31 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 37 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 43 | \( 1 + (1.16 - 0.183i)T + (0.951 - 0.309i)T^{2} \) |
| 47 | \( 1 + (-0.891 - 0.453i)T^{2} \) |
| 53 | \( 1 + (-0.987 - 0.156i)T^{2} \) |
| 59 | \( 1 + (0.831 - 1.14i)T + (-0.309 - 0.951i)T^{2} \) |
| 61 | \( 1 + (0.951 + 0.309i)T^{2} \) |
| 67 | \( 1 + (1.15 - 0.987i)T + (0.156 - 0.987i)T^{2} \) |
| 71 | \( 1 + (0.156 + 0.987i)T^{2} \) |
| 73 | \( 1 + (-1.14 + 1.14i)T - iT^{2} \) |
| 79 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 83 | \( 1 + 0.618iT - T^{2} \) |
| 89 | \( 1 + (1.26 - 0.303i)T + (0.891 - 0.453i)T^{2} \) |
| 97 | \( 1 + (1.10 + 1.29i)T + (-0.156 + 0.987i)T^{2} \) |
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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
−9.886035446453382622064907744658, −9.380438235023222628243937143046, −8.661739444037617210102447797887, −7.76134268114074386140488148993, −7.05000613287504241153657614474, −5.32934544370062284113212298271, −4.57011380967346997556385629507, −3.37195439493531253720698753761, −2.79447339986914763966568913865, −1.47358229936521453418016369015,
1.60382354151829398447131311954, 3.20769466812265692075035799423, 4.18811297717818459472537249880, 5.15394356460842383090198462359, 6.38852154842926891377296702956, 6.95910885779339952301945504832, 8.121058361115834031758767123571, 8.312173006272035974280826024598, 9.334116942720237311547971602592, 9.994104021486125378569130917397