L(s) = 1 | + (0.755 − 0.654i)3-s + (−0.909 − 0.415i)7-s + (0.142 − 0.989i)9-s + (−0.959 − 0.281i)11-s + (−0.909 + 0.415i)13-s + (0.540 − 0.841i)17-s + (−0.841 + 0.540i)19-s + (−0.959 + 0.281i)21-s + (−0.540 − 0.841i)27-s + (0.841 + 0.540i)29-s + (−0.654 + 0.755i)31-s + (−0.909 + 0.415i)33-s + (−0.989 − 0.142i)37-s + (−0.415 + 0.909i)39-s + (−0.142 − 0.989i)41-s + ⋯ |
L(s) = 1 | + (0.755 − 0.654i)3-s + (−0.909 − 0.415i)7-s + (0.142 − 0.989i)9-s + (−0.959 − 0.281i)11-s + (−0.909 + 0.415i)13-s + (0.540 − 0.841i)17-s + (−0.841 + 0.540i)19-s + (−0.959 + 0.281i)21-s + (−0.540 − 0.841i)27-s + (0.841 + 0.540i)29-s + (−0.654 + 0.755i)31-s + (−0.909 + 0.415i)33-s + (−0.989 − 0.142i)37-s + (−0.415 + 0.909i)39-s + (−0.142 − 0.989i)41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.907 + 0.418i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.907 + 0.418i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.07093418578 - 0.3230306727i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.07093418578 - 0.3230306727i\) |
\(L(1)\) |
\(\approx\) |
\(0.8161701172 - 0.2994146722i\) |
\(L(1)\) |
\(\approx\) |
\(0.8161701172 - 0.2994146722i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 \) |
good | 3 | \( 1 + (0.755 - 0.654i)T \) |
| 7 | \( 1 + (-0.909 - 0.415i)T \) |
| 11 | \( 1 + (-0.959 - 0.281i)T \) |
| 13 | \( 1 + (-0.909 + 0.415i)T \) |
| 17 | \( 1 + (0.540 - 0.841i)T \) |
| 19 | \( 1 + (-0.841 + 0.540i)T \) |
| 29 | \( 1 + (0.841 + 0.540i)T \) |
| 31 | \( 1 + (-0.654 + 0.755i)T \) |
| 37 | \( 1 + (-0.989 - 0.142i)T \) |
| 41 | \( 1 + (-0.142 - 0.989i)T \) |
| 43 | \( 1 + (-0.755 + 0.654i)T \) |
| 47 | \( 1 + iT \) |
| 53 | \( 1 + (-0.909 - 0.415i)T \) |
| 59 | \( 1 + (0.415 + 0.909i)T \) |
| 61 | \( 1 + (-0.654 + 0.755i)T \) |
| 67 | \( 1 + (-0.281 - 0.959i)T \) |
| 71 | \( 1 + (-0.959 + 0.281i)T \) |
| 73 | \( 1 + (-0.540 - 0.841i)T \) |
| 79 | \( 1 + (0.415 + 0.909i)T \) |
| 83 | \( 1 + (0.989 + 0.142i)T \) |
| 89 | \( 1 + (-0.654 - 0.755i)T \) |
| 97 | \( 1 + (-0.989 + 0.142i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−21.99340394115557264428373795792, −21.68684362760669540944451185303, −20.729536378208434382278544589200, −19.96731049593131789140979101043, −19.238408857618370867962331754063, −18.7000317521640783759129452163, −17.45516943908891392353319715427, −16.67699925524783278951157769272, −15.76462587178980832480187728730, −15.20104358729669180013171302182, −14.6062379583244134313305944612, −13.38031728370089822196805100542, −12.90284946171438637441010275565, −11.99448237257381569439516417017, −10.60232410977524258339475625259, −10.13556041568715057797115047379, −9.38547014664572080273849265519, −8.42904791022094843815064229548, −7.73862296824238223369665637650, −6.657789814512237791343641464499, −5.51425172891804148773678619617, −4.72567715792879767770563959284, −3.62697914524166298318098100753, −2.78562988823454313866415537208, −2.05884215841706994482317739721,
0.11789519954353322643899546048, 1.58816268319017730805223826451, 2.74868448348458153426065633704, 3.30150706190089175366436295752, 4.5142002265970751066737488332, 5.689300575009091728155249130982, 6.78335874085749667967058111015, 7.314552607003128861469348233925, 8.22589984237684453394244270152, 9.137869356200991965363848468497, 9.944110616806968949179374031743, 10.71568180523747069980348585515, 12.1685212770708809490716060748, 12.54070457585185033087532119437, 13.492697923016461949754502777966, 14.09446396766656242442416205194, 14.90045824461162518754923655130, 15.93119132762447214028287563640, 16.55778857923696464084491374223, 17.62523080816789861028618015709, 18.44951986929785875150339076008, 19.232895800485267275552428032297, 19.61365442161040570607022665320, 20.64802731576505191348549740218, 21.19917316183236240187372104323