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
−18.530051260386591637744401314771, −17.45945432468282593445548162654, −16.69787713795539758906754419632, −16.25468108406382517888153749196, −15.32930677684092425253993125146, −14.61773558561813177308158824876, −14.14754478349969206644677557271, −13.38009042879128377090733464249, −12.67456054437711410273112866126, −11.75119719350003708764577180914, −11.21969772042947512862522123253, −10.851109699987525692319037951846, −9.87706812113120190830032057504, −9.20350700094420818871481041002, −8.64640151646752248236735131980, −7.99752633774089161994672440978, −7.29653673080147348687222315925, −5.496848953527053462318111594919, −5.22807069038615519525579854016, −4.49409466648898815374561403429, −3.870401278980880091367810663225, −3.23148828592542515024226439526, −2.21793550030832980164987455363, −1.474540209928333470673220107355, −0.22808214711911979871605507106,
0.94180469491077196949181158076, 2.28971304039336422909278887942, 2.83891817118463030268992099195, 3.85523308642816343078258513188, 4.62420538039653640502337954070, 5.52889091107997373769507223808, 6.15496095396502350892690901089, 7.136392325987804248744548596767, 7.53866257358892404916634369584, 8.04003638165059054561381773581, 8.619681741997917414166427553328, 9.49127602267053905424285357241, 10.685977447377547425400304312326, 11.120647471450660641094048330464, 12.297066194832186351979957813592, 12.70603089456523331334874318043, 13.2697283135839177738043202931, 14.37052183416860151652648799363, 14.62799966432115443748455389990, 15.33649511768199924371402866417, 15.57362523688352986283873571428, 17.002661075116574163718719142031, 17.39362742037038297579030915505, 18.19879331796327087745717802568, 18.426596520376918881687668470154