L(s) = 1 | + (0.601 − 0.573i)2-s + (−0.0147 + 0.309i)4-s + (0.235 + 0.971i)7-s + (0.712 + 0.822i)8-s + (−0.959 − 0.281i)9-s + (0.839 − 1.17i)11-s + (0.698 + 0.449i)14-s + (0.591 + 0.0564i)16-s + (−0.738 + 0.380i)18-s + (−0.171 − 1.19i)22-s + (−1.54 − 1.21i)23-s + (−0.654 + 0.755i)25-s + (−0.304 + 0.0586i)28-s + (−0.235 − 0.408i)29-s + (−0.467 + 0.367i)32-s + ⋯ |
L(s) = 1 | + (0.601 − 0.573i)2-s + (−0.0147 + 0.309i)4-s + (0.235 + 0.971i)7-s + (0.712 + 0.822i)8-s + (−0.959 − 0.281i)9-s + (0.839 − 1.17i)11-s + (0.698 + 0.449i)14-s + (0.591 + 0.0564i)16-s + (−0.738 + 0.380i)18-s + (−0.171 − 1.19i)22-s + (−1.54 − 1.21i)23-s + (−0.654 + 0.755i)25-s + (−0.304 + 0.0586i)28-s + (−0.235 − 0.408i)29-s + (−0.467 + 0.367i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 469 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.133i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 469 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.133i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.154000104\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.154000104\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (-0.235 - 0.971i)T \) |
| 67 | \( 1 + (0.995 + 0.0950i)T \) |
good | 2 | \( 1 + (-0.601 + 0.573i)T + (0.0475 - 0.998i)T^{2} \) |
| 3 | \( 1 + (0.959 + 0.281i)T^{2} \) |
| 5 | \( 1 + (0.654 - 0.755i)T^{2} \) |
| 11 | \( 1 + (-0.839 + 1.17i)T + (-0.327 - 0.945i)T^{2} \) |
| 13 | \( 1 + (0.786 - 0.618i)T^{2} \) |
| 17 | \( 1 + (0.995 - 0.0950i)T^{2} \) |
| 19 | \( 1 + (0.888 + 0.458i)T^{2} \) |
| 23 | \( 1 + (1.54 + 1.21i)T + (0.235 + 0.971i)T^{2} \) |
| 29 | \( 1 + (0.235 + 0.408i)T + (-0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (0.786 + 0.618i)T^{2} \) |
| 37 | \( 1 + (-0.995 + 1.72i)T + (-0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + (-0.580 - 0.814i)T^{2} \) |
| 43 | \( 1 + (1.49 - 0.961i)T + (0.415 - 0.909i)T^{2} \) |
| 47 | \( 1 + (-0.723 + 0.690i)T^{2} \) |
| 53 | \( 1 + (-1.56 - 1.00i)T + (0.415 + 0.909i)T^{2} \) |
| 59 | \( 1 + (0.142 - 0.989i)T^{2} \) |
| 61 | \( 1 + (0.327 - 0.945i)T^{2} \) |
| 71 | \( 1 + (-0.0552 + 1.15i)T + (-0.995 - 0.0950i)T^{2} \) |
| 73 | \( 1 + (0.327 - 0.945i)T^{2} \) |
| 79 | \( 1 + (0.279 + 0.0538i)T + (0.928 + 0.371i)T^{2} \) |
| 83 | \( 1 + (-0.981 - 0.189i)T^{2} \) |
| 89 | \( 1 + (0.959 - 0.281i)T^{2} \) |
| 97 | \( 1 + (0.5 + 0.866i)T^{2} \) |
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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
−11.71074437833429849684656836164, −10.71489716607468370186032851723, −9.271916291901708822350470296698, −8.561646198050393937524448830341, −7.85356669730853126808871477794, −6.19415354004797674062295759305, −5.60119507903777712689846378850, −4.22062157233810351768440088206, −3.24213539208999837288510726343, −2.19395926388987145533005302223,
1.74405819130659089119328917482, 3.76359856007305651369530779351, 4.58707854820947426147429108928, 5.61975580407820207276097285724, 6.58370320472570887412285613814, 7.38968959957527476503277095788, 8.335423228986523885459055611883, 9.796977467522524207958921763864, 10.13931312899715512325206394959, 11.39657339358862320314522834364