L(s) = 1 | + (−0.142 − 0.989i)3-s + (0.755 + 0.654i)7-s + (−0.959 + 0.281i)9-s + (0.540 + 0.841i)11-s + (0.654 + 0.755i)13-s + (−0.909 − 0.415i)17-s + (−0.909 + 0.415i)19-s + (0.540 − 0.841i)21-s + (0.415 + 0.909i)27-s + (−0.909 − 0.415i)29-s + (0.142 − 0.989i)31-s + (0.755 − 0.654i)33-s + (0.959 − 0.281i)37-s + (0.654 − 0.755i)39-s + (0.959 + 0.281i)41-s + ⋯ |
L(s) = 1 | + (−0.142 − 0.989i)3-s + (0.755 + 0.654i)7-s + (−0.959 + 0.281i)9-s + (0.540 + 0.841i)11-s + (0.654 + 0.755i)13-s + (−0.909 − 0.415i)17-s + (−0.909 + 0.415i)19-s + (0.540 − 0.841i)21-s + (0.415 + 0.909i)27-s + (−0.909 − 0.415i)29-s + (0.142 − 0.989i)31-s + (0.755 − 0.654i)33-s + (0.959 − 0.281i)37-s + (0.654 − 0.755i)39-s + (0.959 + 0.281i)41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.536 + 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.536 + 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.098399096 + 0.6033374419i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.098399096 + 0.6033374419i\) |
\(L(1)\) |
\(\approx\) |
\(1.012436602 + 0.02455697790i\) |
\(L(1)\) |
\(\approx\) |
\(1.012436602 + 0.02455697790i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 \) |
good | 3 | \( 1 + (-0.142 - 0.989i)T \) |
| 7 | \( 1 + (0.755 + 0.654i)T \) |
| 11 | \( 1 + (0.540 + 0.841i)T \) |
| 13 | \( 1 + (0.654 + 0.755i)T \) |
| 17 | \( 1 + (-0.909 - 0.415i)T \) |
| 19 | \( 1 + (-0.909 + 0.415i)T \) |
| 29 | \( 1 + (-0.909 - 0.415i)T \) |
| 31 | \( 1 + (0.142 - 0.989i)T \) |
| 37 | \( 1 + (0.959 - 0.281i)T \) |
| 41 | \( 1 + (0.959 + 0.281i)T \) |
| 43 | \( 1 + (0.142 + 0.989i)T \) |
| 47 | \( 1 - iT \) |
| 53 | \( 1 + (-0.654 + 0.755i)T \) |
| 59 | \( 1 + (-0.755 + 0.654i)T \) |
| 61 | \( 1 + (-0.989 - 0.142i)T \) |
| 67 | \( 1 + (-0.841 - 0.540i)T \) |
| 71 | \( 1 + (0.841 + 0.540i)T \) |
| 73 | \( 1 + (0.909 - 0.415i)T \) |
| 79 | \( 1 + (-0.654 - 0.755i)T \) |
| 83 | \( 1 + (-0.959 + 0.281i)T \) |
| 89 | \( 1 + (-0.142 - 0.989i)T \) |
| 97 | \( 1 + (-0.281 + 0.959i)T \) |
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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
−20.03097749030297511403890716317, −19.62844788015548716767152883633, −18.42634856559222530761497165708, −17.62127761749030128299236728453, −17.05332288351732584429274357230, −16.42948496925203107766871410870, −15.5365675422792983375332762018, −14.97918794523417164436702222463, −14.18724125757366809386953164167, −13.52878644562649912516841817933, −12.61058552049691254886552502368, −11.44307621082882417068410658697, −10.90222592102842740157097259850, −10.59503030023695794987856648006, −9.4630835137077072290693518442, −8.608650661247312415284551003152, −8.233829929385060201176580959981, −6.96173030639143038572955897591, −6.090631406378572959862538326737, −5.32380149983370653016373498661, −4.38243695860806398996400770554, −3.83558746579090302305207398464, −2.957091269711755526159954460775, −1.6790463308928148183210110072, −0.4571188513207390902832367816,
1.25460019496141528671131474284, 1.98842939595512844026246375549, 2.62036825858218766399108128478, 4.1400684874368406287824413661, 4.727258199778463375704734274916, 6.08840630885384168909871309673, 6.25269875615768890781137783251, 7.44929562343763776215026417935, 7.956083439267052922922704718419, 8.985314526745869082995939816715, 9.3981290176940685323911083080, 10.95673848117019541432391809389, 11.31792785878126420495583181457, 12.09599860771493883339611676502, 12.78969320501041937071700890051, 13.51462385262284915778151063715, 14.41130042903495549758762794797, 14.885395033199123556762935386630, 15.81896769300873038864454769551, 16.88401590707662019508303134492, 17.37610976825644913081701088500, 18.25123773354898201529690389036, 18.572485144972391966575420013332, 19.45312606148781789975083653661, 20.17222252236919456285274150425