| L(s) = 1 | + (0.142 − 0.989i)2-s + (−0.333 − 0.384i)3-s + (−0.959 − 0.281i)4-s + (0.841 + 0.540i)5-s + (−0.428 + 0.275i)6-s + (−0.617 + 4.29i)7-s + (−0.415 + 0.909i)8-s + (0.390 − 2.71i)9-s + (0.654 − 0.755i)10-s + (−2.08 − 1.34i)11-s + (0.211 + 0.463i)12-s + (1.18 + 2.58i)13-s + (4.16 + 1.22i)14-s + (−0.0724 − 0.504i)15-s + (0.841 + 0.540i)16-s + (6.50 − 1.91i)17-s + ⋯ |
| L(s) = 1 | + (0.100 − 0.699i)2-s + (−0.192 − 0.222i)3-s + (−0.479 − 0.140i)4-s + (0.376 + 0.241i)5-s + (−0.174 + 0.112i)6-s + (−0.233 + 1.62i)7-s + (−0.146 + 0.321i)8-s + (0.130 − 0.904i)9-s + (0.207 − 0.238i)10-s + (−0.628 − 0.404i)11-s + (0.0610 + 0.133i)12-s + (0.327 + 0.718i)13-s + (1.11 + 0.326i)14-s + (−0.0187 − 0.130i)15-s + (0.210 + 0.135i)16-s + (1.57 − 0.463i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.119i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 + 0.119i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.45048 - 0.0867281i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.45048 - 0.0867281i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.142 + 0.989i)T \) |
| 5 | \( 1 + (-0.841 - 0.540i)T \) |
| 67 | \( 1 + (3.40 + 7.44i)T \) |
| good | 3 | \( 1 + (0.333 + 0.384i)T + (-0.426 + 2.96i)T^{2} \) |
| 7 | \( 1 + (0.617 - 4.29i)T + (-6.71 - 1.97i)T^{2} \) |
| 11 | \( 1 + (2.08 + 1.34i)T + (4.56 + 10.0i)T^{2} \) |
| 13 | \( 1 + (-1.18 - 2.58i)T + (-8.51 + 9.82i)T^{2} \) |
| 17 | \( 1 + (-6.50 + 1.91i)T + (14.3 - 9.19i)T^{2} \) |
| 19 | \( 1 + (-0.629 - 4.38i)T + (-18.2 + 5.35i)T^{2} \) |
| 23 | \( 1 + (-4.17 - 4.81i)T + (-3.27 + 22.7i)T^{2} \) |
| 29 | \( 1 - 8.42T + 29T^{2} \) |
| 31 | \( 1 + (1.60 - 3.51i)T + (-20.3 - 23.4i)T^{2} \) |
| 37 | \( 1 + 4.35T + 37T^{2} \) |
| 41 | \( 1 + (1.51 - 0.445i)T + (34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (-10.7 + 3.15i)T + (36.1 - 23.2i)T^{2} \) |
| 47 | \( 1 + (2.92 + 3.37i)T + (-6.68 + 46.5i)T^{2} \) |
| 53 | \( 1 + (-1.24 - 0.364i)T + (44.5 + 28.6i)T^{2} \) |
| 59 | \( 1 + (-1.08 + 2.37i)T + (-38.6 - 44.5i)T^{2} \) |
| 61 | \( 1 + (1.28 - 0.823i)T + (25.3 - 55.4i)T^{2} \) |
| 71 | \( 1 + (8.50 + 2.49i)T + (59.7 + 38.3i)T^{2} \) |
| 73 | \( 1 + (-4.76 + 3.06i)T + (30.3 - 66.4i)T^{2} \) |
| 79 | \( 1 + (-6.84 - 14.9i)T + (-51.7 + 59.7i)T^{2} \) |
| 83 | \( 1 + (-1.66 - 1.06i)T + (34.4 + 75.4i)T^{2} \) |
| 89 | \( 1 + (11.0 - 12.7i)T + (-12.6 - 88.0i)T^{2} \) |
| 97 | \( 1 + 15.0T + 97T^{2} \) |
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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
−10.49510665134076914351378157160, −9.579110907619199731885705864332, −9.063470101714677753849811091246, −8.091415133284524790913511099382, −6.76048180616060573338623773282, −5.75429905482146808748138582568, −5.28801545731120518938940763719, −3.53251079519001449832074258362, −2.77816165145826033960354733727, −1.38926036154625323523744430436,
0.903952793656405027072272479941, 2.98544217860313491980823804062, 4.34111619047705514153182919494, 5.01072375451520343488837022974, 6.00279946322148596247989858493, 7.16800315620093717507379628892, 7.69088567421520872389856966852, 8.591767606120885485522683628690, 9.954284187233624721576556636020, 10.32286471530244672721851366577
