| L(s) = 1 | + 2-s − 4-s − 3·8-s − 4·11-s − 16-s + 2·17-s − 4·19-s − 4·22-s + 2·29-s + 5·32-s + 2·34-s − 10·37-s − 4·38-s + 10·41-s − 4·43-s + 4·44-s − 8·47-s − 7·49-s − 10·53-s + 2·58-s − 4·59-s − 2·61-s + 7·64-s + 12·67-s − 2·68-s − 8·71-s + 10·73-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 1.06·8-s − 1.20·11-s − 1/4·16-s + 0.485·17-s − 0.917·19-s − 0.852·22-s + 0.371·29-s + 0.883·32-s + 0.342·34-s − 1.64·37-s − 0.648·38-s + 1.56·41-s − 0.609·43-s + 0.603·44-s − 1.16·47-s − 49-s − 1.37·53-s + 0.262·58-s − 0.520·59-s − 0.256·61-s + 7/8·64-s + 1.46·67-s − 0.242·68-s − 0.949·71-s + 1.17·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9144742382\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9144742382\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−14.60739697870665, −14.38850645149216, −13.79401305822607, −13.20543771007693, −12.85930146219388, −12.39978505114474, −11.95755657371995, −11.04258370204017, −10.82170228675921, −9.945231596659135, −9.703424120049263, −8.945275556481718, −8.225714697200211, −8.112433309641856, −7.219685386680302, −6.527312553762123, −5.980103712600091, −5.349226168081437, −4.880321890804477, −4.391654426292730, −3.574710518392486, −3.082707304674841, −2.403870979514865, −1.496796016303638, −0.3092222087661859,
0.3092222087661859, 1.496796016303638, 2.403870979514865, 3.082707304674841, 3.574710518392486, 4.391654426292730, 4.880321890804477, 5.349226168081437, 5.980103712600091, 6.527312553762123, 7.219685386680302, 8.112433309641856, 8.225714697200211, 8.945275556481718, 9.703424120049263, 9.945231596659135, 10.82170228675921, 11.04258370204017, 11.95755657371995, 12.39978505114474, 12.85930146219388, 13.20543771007693, 13.79401305822607, 14.38850645149216, 14.60739697870665
