L(s) = 1 | + (0.142 − 1.98i)3-s + (−0.654 + 0.755i)4-s + (0.368 − 1.25i)5-s + (−2.94 − 0.424i)9-s + (−0.959 + 0.281i)11-s + (1.41 + 1.41i)12-s + (−2.44 − 0.913i)15-s + (−0.142 − 0.989i)16-s + (0.708 + 1.10i)20-s + (1.40 − 1.05i)23-s + (−0.601 − 0.386i)25-s + (−0.839 + 3.85i)27-s + (−0.559 − 0.418i)31-s + (0.424 + 1.94i)33-s + (2.25 − 1.95i)36-s + (−0.100 + 0.100i)37-s + ⋯ |
L(s) = 1 | + (0.142 − 1.98i)3-s + (−0.654 + 0.755i)4-s + (0.368 − 1.25i)5-s + (−2.94 − 0.424i)9-s + (−0.959 + 0.281i)11-s + (1.41 + 1.41i)12-s + (−2.44 − 0.913i)15-s + (−0.142 − 0.989i)16-s + (0.708 + 1.10i)20-s + (1.40 − 1.05i)23-s + (−0.601 − 0.386i)25-s + (−0.839 + 3.85i)27-s + (−0.559 − 0.418i)31-s + (0.424 + 1.94i)33-s + (2.25 − 1.95i)36-s + (−0.100 + 0.100i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 979 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.960 + 0.279i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 979 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.960 + 0.279i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7626828256\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7626828256\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 + (0.959 - 0.281i)T \) |
| 89 | \( 1 + (-0.959 - 0.281i)T \) |
good | 2 | \( 1 + (0.654 - 0.755i)T^{2} \) |
| 3 | \( 1 + (-0.142 + 1.98i)T + (-0.989 - 0.142i)T^{2} \) |
| 5 | \( 1 + (-0.368 + 1.25i)T + (-0.841 - 0.540i)T^{2} \) |
| 7 | \( 1 + (0.540 - 0.841i)T^{2} \) |
| 13 | \( 1 + (-0.989 - 0.142i)T^{2} \) |
| 17 | \( 1 + (-0.654 - 0.755i)T^{2} \) |
| 19 | \( 1 + (0.281 - 0.959i)T^{2} \) |
| 23 | \( 1 + (-1.40 + 1.05i)T + (0.281 - 0.959i)T^{2} \) |
| 29 | \( 1 + (-0.540 + 0.841i)T^{2} \) |
| 31 | \( 1 + (0.559 + 0.418i)T + (0.281 + 0.959i)T^{2} \) |
| 37 | \( 1 + (0.100 - 0.100i)T - iT^{2} \) |
| 41 | \( 1 + (-0.989 + 0.142i)T^{2} \) |
| 43 | \( 1 + (-0.540 - 0.841i)T^{2} \) |
| 47 | \( 1 + (-0.817 - 0.708i)T + (0.142 + 0.989i)T^{2} \) |
| 53 | \( 1 + (-1.45 + 1.25i)T + (0.142 - 0.989i)T^{2} \) |
| 59 | \( 1 + (0.114 + 1.59i)T + (-0.989 + 0.142i)T^{2} \) |
| 61 | \( 1 + (-0.909 - 0.415i)T^{2} \) |
| 67 | \( 1 + (1.19 + 1.37i)T + (-0.142 + 0.989i)T^{2} \) |
| 71 | \( 1 + (0.234 + 0.797i)T + (-0.841 + 0.540i)T^{2} \) |
| 73 | \( 1 + (0.959 - 0.281i)T^{2} \) |
| 79 | \( 1 + (-0.959 + 0.281i)T^{2} \) |
| 83 | \( 1 + (-0.755 + 0.654i)T^{2} \) |
| 97 | \( 1 + (-0.540 - 0.158i)T + (0.841 + 0.540i)T^{2} \) |
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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
−9.287506033947646344508019165171, −8.810322239403507944662592895728, −8.066020691371670073986937607504, −7.53934798890945252130701029708, −6.60032179312237340603466351178, −5.47948660540364415824842246514, −4.78660319007787815440190125377, −3.11870134805717224125460904058, −2.09626202649743980517501277663, −0.71801525204924389909794120623,
2.63914770381117364588221776782, 3.43646471085157167704185819408, 4.44621449779154143691741818200, 5.45034348144385774367537325695, 5.76015387388973334717068978482, 7.18236503953550723750083748407, 8.557450215503435294689405307513, 9.144801110449918434836448000943, 9.947946442175962225378261240263, 10.53138038403417072180461211166