| L(s) = 1 | + (0.984 + 0.173i)5-s + (−0.984 + 0.173i)11-s + (0.642 − 0.766i)13-s + 17-s − i·19-s + (−0.766 − 0.642i)23-s + (0.939 + 0.342i)25-s + (−0.642 − 0.766i)29-s + (−0.939 + 0.342i)31-s + (0.866 − 0.5i)37-s + (−0.766 − 0.642i)41-s + (0.342 − 0.939i)43-s + (−0.939 − 0.342i)47-s + (0.866 − 0.5i)53-s − 55-s + ⋯ |
| L(s) = 1 | + (0.984 + 0.173i)5-s + (−0.984 + 0.173i)11-s + (0.642 − 0.766i)13-s + 17-s − i·19-s + (−0.766 − 0.642i)23-s + (0.939 + 0.342i)25-s + (−0.642 − 0.766i)29-s + (−0.939 + 0.342i)31-s + (0.866 − 0.5i)37-s + (−0.766 − 0.642i)41-s + (0.342 − 0.939i)43-s + (−0.939 − 0.342i)47-s + (0.866 − 0.5i)53-s − 55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.00934 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.00934 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.117616244 - 1.107218569i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.117616244 - 1.107218569i\) |
| \(L(1)\) |
\(\approx\) |
\(1.142153190 - 0.1925813117i\) |
| \(L(1)\) |
\(\approx\) |
\(1.142153190 - 0.1925813117i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + (0.984 + 0.173i)T \) |
| 11 | \( 1 + (-0.984 + 0.173i)T \) |
| 13 | \( 1 + (0.642 - 0.766i)T \) |
| 17 | \( 1 + T \) |
| 19 | \( 1 - iT \) |
| 23 | \( 1 + (-0.766 - 0.642i)T \) |
| 29 | \( 1 + (-0.642 - 0.766i)T \) |
| 31 | \( 1 + (-0.939 + 0.342i)T \) |
| 37 | \( 1 + (0.866 - 0.5i)T \) |
| 41 | \( 1 + (-0.766 - 0.642i)T \) |
| 43 | \( 1 + (0.342 - 0.939i)T \) |
| 47 | \( 1 + (-0.939 - 0.342i)T \) |
| 53 | \( 1 + (0.866 - 0.5i)T \) |
| 59 | \( 1 + (-0.642 + 0.766i)T \) |
| 61 | \( 1 + (-0.342 + 0.939i)T \) |
| 67 | \( 1 + (-0.984 - 0.173i)T \) |
| 71 | \( 1 + (0.5 - 0.866i)T \) |
| 73 | \( 1 + (0.5 - 0.866i)T \) |
| 79 | \( 1 + (0.173 + 0.984i)T \) |
| 83 | \( 1 + (-0.642 - 0.766i)T \) |
| 89 | \( 1 - T \) |
| 97 | \( 1 + (-0.939 - 0.342i)T \) |
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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
−18.99372713029523862190166949941, −18.32775673474467550186129920133, −18.14011387638597561452325290516, −16.94531866557822219478122368839, −16.5202840158649395145846111843, −15.94724997789399543031093206832, −14.85954159420774835926113392858, −14.267949318691351805979712466597, −13.60009970907726421901175128810, −12.96798792349252476359881053075, −12.31432139021442752063863743371, −11.33830384652736982836269605587, −10.6912103147407801296367406863, −9.77234777461182004374037004557, −9.51093635644009081982587690269, −8.37887328685100850706416005545, −7.86098669404904963498397043872, −6.87754186509611853918836503948, −5.89706087915570038450877620963, −5.624204203858482879004133770411, −4.67326475686741769422694456153, −3.66035365946042011257576583251, −2.8826025502802435186734036252, −1.81304956369157819642108519711, −1.30440759651916807533659794512,
0.44843832207378915439090710275, 1.63828875846291974059668516374, 2.4625545322864351396859638105, 3.1493259005665817521666671603, 4.15144362616576378064485180393, 5.3412153356454806225343459044, 5.56478085079630106214679252634, 6.47671452286750039709110917635, 7.37018701998754110345440021102, 8.06342801794004340339407913153, 8.92673607461347775913539691395, 9.67006208709211897282702808292, 10.480699343285843695172271971437, 10.742357634929256412395578525276, 11.86500800860522442658385648570, 12.70860107034431663798100219488, 13.31707797916198964254810434686, 13.7939801696693940553816212670, 14.75095550578812269231616724706, 15.27780312854453784348102989641, 16.170320497296622964565433948212, 16.78462441894104854609958233985, 17.61449016899361512296265695817, 18.325290437291974256185885611525, 18.46477515579540948692534063746
