L(s) = 1 | + (0.173 − 0.984i)2-s + (0.396 + 0.918i)3-s + (−0.939 − 0.342i)4-s + (−0.686 − 0.727i)5-s + (0.973 − 0.230i)6-s + (0.973 − 0.230i)7-s + (−0.5 + 0.866i)8-s + (−0.686 + 0.727i)9-s + (−0.835 + 0.549i)10-s + (−0.835 − 0.549i)11-s + (−0.0581 − 0.998i)12-s + (−0.286 + 0.957i)13-s + (−0.0581 − 0.998i)14-s + (0.396 − 0.918i)15-s + (0.766 + 0.642i)16-s + (0.173 − 0.984i)17-s + ⋯ |
L(s) = 1 | + (0.173 − 0.984i)2-s + (0.396 + 0.918i)3-s + (−0.939 − 0.342i)4-s + (−0.686 − 0.727i)5-s + (0.973 − 0.230i)6-s + (0.973 − 0.230i)7-s + (−0.5 + 0.866i)8-s + (−0.686 + 0.727i)9-s + (−0.835 + 0.549i)10-s + (−0.835 − 0.549i)11-s + (−0.0581 − 0.998i)12-s + (−0.286 + 0.957i)13-s + (−0.0581 − 0.998i)14-s + (0.396 − 0.918i)15-s + (0.766 + 0.642i)16-s + (0.173 − 0.984i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.890 - 0.455i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.890 - 0.455i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1837513453 - 0.7631348154i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1837513453 - 0.7631348154i\) |
\(L(1)\) |
\(\approx\) |
\(0.8593525365 - 0.3330260107i\) |
\(L(1)\) |
\(\approx\) |
\(0.8593525365 - 0.3330260107i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 37 | \( 1 \) |
| 109 | \( 1 \) |
good | 2 | \( 1 + (0.173 - 0.984i)T \) |
| 3 | \( 1 + (0.396 + 0.918i)T \) |
| 5 | \( 1 + (-0.686 - 0.727i)T \) |
| 7 | \( 1 + (0.973 - 0.230i)T \) |
| 11 | \( 1 + (-0.835 - 0.549i)T \) |
| 13 | \( 1 + (-0.286 + 0.957i)T \) |
| 17 | \( 1 + (0.173 - 0.984i)T \) |
| 19 | \( 1 + (-0.5 + 0.866i)T \) |
| 23 | \( 1 + (0.766 + 0.642i)T \) |
| 29 | \( 1 + (-0.835 + 0.549i)T \) |
| 31 | \( 1 + (-0.993 + 0.116i)T \) |
| 41 | \( 1 + (0.766 + 0.642i)T \) |
| 43 | \( 1 + (0.766 - 0.642i)T \) |
| 47 | \( 1 + (-0.286 - 0.957i)T \) |
| 53 | \( 1 + (-0.0581 - 0.998i)T \) |
| 59 | \( 1 + (0.396 - 0.918i)T \) |
| 61 | \( 1 + (-0.835 + 0.549i)T \) |
| 67 | \( 1 + (0.893 - 0.448i)T \) |
| 71 | \( 1 + (0.173 + 0.984i)T \) |
| 73 | \( 1 + (0.893 - 0.448i)T \) |
| 79 | \( 1 + (-0.835 - 0.549i)T \) |
| 83 | \( 1 + (-0.686 - 0.727i)T \) |
| 89 | \( 1 + (-0.993 - 0.116i)T \) |
| 97 | \( 1 + (0.396 + 0.918i)T \) |
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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
−18.426596520376918881687668470154, −18.19879331796327087745717802568, −17.39362742037038297579030915505, −17.002661075116574163718719142031, −15.57362523688352986283873571428, −15.33649511768199924371402866417, −14.62799966432115443748455389990, −14.37052183416860151652648799363, −13.2697283135839177738043202931, −12.70603089456523331334874318043, −12.297066194832186351979957813592, −11.120647471450660641094048330464, −10.685977447377547425400304312326, −9.49127602267053905424285357241, −8.619681741997917414166427553328, −8.04003638165059054561381773581, −7.53866257358892404916634369584, −7.136392325987804248744548596767, −6.15496095396502350892690901089, −5.52889091107997373769507223808, −4.62420538039653640502337954070, −3.85523308642816343078258513188, −2.83891817118463030268992099195, −2.28971304039336422909278887942, −0.94180469491077196949181158076,
0.22808214711911979871605507106, 1.474540209928333470673220107355, 2.21793550030832980164987455363, 3.23148828592542515024226439526, 3.870401278980880091367810663225, 4.49409466648898815374561403429, 5.22807069038615519525579854016, 5.496848953527053462318111594919, 7.29653673080147348687222315925, 7.99752633774089161994672440978, 8.64640151646752248236735131980, 9.20350700094420818871481041002, 9.87706812113120190830032057504, 10.851109699987525692319037951846, 11.21969772042947512862522123253, 11.75119719350003708764577180914, 12.67456054437711410273112866126, 13.38009042879128377090733464249, 14.14754478349969206644677557271, 14.61773558561813177308158824876, 15.32930677684092425253993125146, 16.25468108406382517888153749196, 16.69787713795539758906754419632, 17.45945432468282593445548162654, 18.530051260386591637744401314771