L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 2·7-s − 8-s + 9-s − 11-s − 12-s + 13-s + 2·14-s + 16-s − 17-s − 18-s + 6·19-s + 2·21-s + 22-s + 24-s − 26-s − 27-s − 2·28-s − 10·29-s − 32-s + 33-s + 34-s + 36-s − 2·37-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.755·7-s − 0.353·8-s + 1/3·9-s − 0.301·11-s − 0.288·12-s + 0.277·13-s + 0.534·14-s + 1/4·16-s − 0.242·17-s − 0.235·18-s + 1.37·19-s + 0.436·21-s + 0.213·22-s + 0.204·24-s − 0.196·26-s − 0.192·27-s − 0.377·28-s − 1.85·29-s − 0.176·32-s + 0.174·33-s + 0.171·34-s + 1/6·36-s − 0.328·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 364650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 364650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 + T \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−12.77983890505252, −12.17533144852196, −11.68838913450432, −11.38234009950435, −10.91819495372463, −10.44732769505005, −10.01524220076928, −9.504255183232875, −9.266149930196073, −8.790924276380547, −8.046352297396485, −7.729337328079491, −7.099039921674955, −6.980589394201930, −6.131617867918499, −5.907534792414991, −5.443876536933883, −4.811541627144498, −4.266243504354685, −3.512633888546915, −3.173805386517194, −2.638237027156910, −1.693392013871544, −1.475539962317269, −0.5356391645936809, 0,
0.5356391645936809, 1.475539962317269, 1.693392013871544, 2.638237027156910, 3.173805386517194, 3.512633888546915, 4.266243504354685, 4.811541627144498, 5.443876536933883, 5.907534792414991, 6.131617867918499, 6.980589394201930, 7.099039921674955, 7.729337328079491, 8.046352297396485, 8.790924276380547, 9.266149930196073, 9.504255183232875, 10.01524220076928, 10.44732769505005, 10.91819495372463, 11.38234009950435, 11.68838913450432, 12.17533144852196, 12.77983890505252