L(s) = 1 | − 2-s + (−0.993 − 0.116i)3-s + 4-s + (0.686 + 0.727i)5-s + (0.993 + 0.116i)6-s + (0.973 − 0.230i)7-s − 8-s + (0.973 + 0.230i)9-s + (−0.686 − 0.727i)10-s + (−0.686 + 0.727i)11-s + (−0.993 − 0.116i)12-s + (0.686 + 0.727i)13-s + (−0.973 + 0.230i)14-s + (−0.597 − 0.802i)15-s + 16-s + (0.5 − 0.866i)17-s + ⋯ |
L(s) = 1 | − 2-s + (−0.993 − 0.116i)3-s + 4-s + (0.686 + 0.727i)5-s + (0.993 + 0.116i)6-s + (0.973 − 0.230i)7-s − 8-s + (0.973 + 0.230i)9-s + (−0.686 − 0.727i)10-s + (−0.686 + 0.727i)11-s + (−0.993 − 0.116i)12-s + (0.686 + 0.727i)13-s + (−0.973 + 0.230i)14-s + (−0.597 − 0.802i)15-s + 16-s + (0.5 − 0.866i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0853 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0853 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6447552941 + 0.7023551544i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6447552941 + 0.7023551544i\) |
\(L(1)\) |
\(\approx\) |
\(0.6632837893 + 0.1645069851i\) |
\(L(1)\) |
\(\approx\) |
\(0.6632837893 + 0.1645069851i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 37 | \( 1 \) |
| 109 | \( 1 \) |
good | 2 | \( 1 - T \) |
| 3 | \( 1 + (-0.993 - 0.116i)T \) |
| 5 | \( 1 + (0.686 + 0.727i)T \) |
| 7 | \( 1 + (0.973 - 0.230i)T \) |
| 11 | \( 1 + (-0.686 + 0.727i)T \) |
| 13 | \( 1 + (0.686 + 0.727i)T \) |
| 17 | \( 1 + (0.5 - 0.866i)T \) |
| 19 | \( 1 + (0.939 - 0.342i)T \) |
| 23 | \( 1 + (-0.173 + 0.984i)T \) |
| 29 | \( 1 + (-0.973 + 0.230i)T \) |
| 31 | \( 1 + (0.0581 - 0.998i)T \) |
| 41 | \( 1 + (0.173 + 0.984i)T \) |
| 43 | \( 1 + (0.939 + 0.342i)T \) |
| 47 | \( 1 + (-0.993 + 0.116i)T \) |
| 53 | \( 1 + (0.396 + 0.918i)T \) |
| 59 | \( 1 + (-0.597 - 0.802i)T \) |
| 61 | \( 1 + (-0.893 - 0.448i)T \) |
| 67 | \( 1 + (-0.993 + 0.116i)T \) |
| 71 | \( 1 + T \) |
| 73 | \( 1 + (0.973 + 0.230i)T \) |
| 79 | \( 1 + (-0.597 - 0.802i)T \) |
| 83 | \( 1 + (-0.835 + 0.549i)T \) |
| 89 | \( 1 + (0.286 - 0.957i)T \) |
| 97 | \( 1 + (0.0581 + 0.998i)T \) |
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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
−18.22134557479008574081569394299, −17.68764350938508364166921743641, −16.962802646891278657223185414794, −16.48077758353043592106753848880, −15.8750983404001431809119045524, −15.21340154152239944280500203509, −14.281141081106930984818250891342, −13.362173360520397098881309483002, −12.479168529608333982122234765, −12.11488962764556623588554288867, −11.11566408049524872328691064026, −10.68235255228187023366645231786, −10.15589600663302519570446929444, −9.29746063687605643660882340989, −8.44213342591296999661338412507, −8.07366761552890415976841145276, −7.18393540531386400126542942827, −6.08612599890367689105028940555, −5.638730902000529015532556217632, −5.240121627590849321858528885160, −4.077394464989298815743287798584, −2.9974989393827371929827478313, −1.842283266770104891783150494958, −1.29283176770117969629235622375, −0.48886554332486067059672860874,
1.06429430478032474214257515696, 1.66641032107972039053181912593, 2.3953814237513392751512283791, 3.459069692201219622262601662179, 4.66063639274330354968310105301, 5.44715972023637473088552239152, 6.023509993707631686776125696505, 6.894570577886592640460243111520, 7.53824443663095969708053926778, 7.82558756698802734847264462050, 9.37023927796073841114235564487, 9.5709488276282195604582988416, 10.42668462423129441457878638018, 11.20521855624626352561689043976, 11.340121868431780495470916857450, 12.14089224515705461351951937173, 13.18964243922912386532571859633, 13.85878373951446339299838419434, 14.73508348821299181192856774033, 15.46372135008931753393096413527, 16.105639677878257376921704402569, 16.85834769701908060637292136078, 17.458103715389573632194258900989, 18.01173505787499843383858923347, 18.44838122969047349898641556808