L(s) = 1 | + (−0.327 + 0.945i)2-s + (0.0395 − 0.0865i)3-s + (−0.786 − 0.618i)4-s + (0.0688 + 0.0656i)6-s + (0.841 − 0.540i)8-s + (0.648 + 0.748i)9-s + (1.21 − 1.16i)11-s + (−0.0845 + 0.0436i)12-s + (0.235 + 0.971i)16-s + (−1.45 + 1.14i)17-s + (−0.919 + 0.368i)18-s + (0.462 + 0.0892i)19-s + (0.698 + 1.53i)22-s + (−0.0135 − 0.0941i)24-s + (0.841 + 0.540i)25-s + ⋯ |
L(s) = 1 | + (−0.327 + 0.945i)2-s + (0.0395 − 0.0865i)3-s + (−0.786 − 0.618i)4-s + (0.0688 + 0.0656i)6-s + (0.841 − 0.540i)8-s + (0.648 + 0.748i)9-s + (1.21 − 1.16i)11-s + (−0.0845 + 0.0436i)12-s + (0.235 + 0.971i)16-s + (−1.45 + 1.14i)17-s + (−0.919 + 0.368i)18-s + (0.462 + 0.0892i)19-s + (0.698 + 1.53i)22-s + (−0.0135 − 0.0941i)24-s + (0.841 + 0.540i)25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.539 - 0.842i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.539 - 0.842i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7783747327\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7783747327\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.327 - 0.945i)T \) |
| 67 | \( 1 + (0.959 + 0.281i)T \) |
good | 3 | \( 1 + (-0.0395 + 0.0865i)T + (-0.654 - 0.755i)T^{2} \) |
| 5 | \( 1 + (-0.841 - 0.540i)T^{2} \) |
| 7 | \( 1 + (-0.928 + 0.371i)T^{2} \) |
| 11 | \( 1 + (-1.21 + 1.16i)T + (0.0475 - 0.998i)T^{2} \) |
| 13 | \( 1 + (0.995 + 0.0950i)T^{2} \) |
| 17 | \( 1 + (1.45 - 1.14i)T + (0.235 - 0.971i)T^{2} \) |
| 19 | \( 1 + (-0.462 - 0.0892i)T + (0.928 + 0.371i)T^{2} \) |
| 23 | \( 1 + (-0.981 + 0.189i)T^{2} \) |
| 29 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (0.995 - 0.0950i)T^{2} \) |
| 37 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + (1.21 + 0.486i)T + (0.723 + 0.690i)T^{2} \) |
| 43 | \( 1 + (0.205 + 1.43i)T + (-0.959 + 0.281i)T^{2} \) |
| 47 | \( 1 + (0.327 - 0.945i)T^{2} \) |
| 53 | \( 1 + (0.959 + 0.281i)T^{2} \) |
| 59 | \( 1 + (1.49 - 0.961i)T + (0.415 - 0.909i)T^{2} \) |
| 61 | \( 1 + (-0.0475 - 0.998i)T^{2} \) |
| 71 | \( 1 + (-0.235 - 0.971i)T^{2} \) |
| 73 | \( 1 + (1.13 + 1.08i)T + (0.0475 + 0.998i)T^{2} \) |
| 79 | \( 1 + (-0.580 + 0.814i)T^{2} \) |
| 83 | \( 1 + (0.452 + 1.86i)T + (-0.888 + 0.458i)T^{2} \) |
| 89 | \( 1 + (0.271 + 0.595i)T + (-0.654 + 0.755i)T^{2} \) |
| 97 | \( 1 + (-0.786 - 1.36i)T + (-0.5 + 0.866i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.86829446931286942902845681604, −10.32432908238191490544795429345, −8.924974624162640347282830430688, −8.728646913997560056919080487925, −7.50860921096591077398211806769, −6.69294082642769237934160408530, −5.88571163108810616532870743301, −4.71678743691970897790020736100, −3.71584161582107853069435804230, −1.58151440606263223617628536540,
1.44849931297916636453214105704, 2.86608013324098290884031516426, 4.18829502873937277372554100223, 4.75168784868128084964356707600, 6.62281750360340739944053354935, 7.25459964089033471729857641194, 8.627693724883521559266860593875, 9.423594269421380526699330264439, 9.808305829126363559558629359406, 10.96009578207721548822514061368