L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 5·7-s + 8-s − 2·9-s − 2·11-s + 12-s + 4·13-s + 5·14-s + 16-s − 17-s − 2·18-s − 7·19-s + 5·21-s − 2·22-s + 2·23-s + 24-s + 4·26-s − 5·27-s + 5·28-s − 3·29-s − 4·31-s + 32-s − 2·33-s − 34-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 1.88·7-s + 0.353·8-s − 2/3·9-s − 0.603·11-s + 0.288·12-s + 1.10·13-s + 1.33·14-s + 1/4·16-s − 0.242·17-s − 0.471·18-s − 1.60·19-s + 1.09·21-s − 0.426·22-s + 0.417·23-s + 0.204·24-s + 0.784·26-s − 0.962·27-s + 0.944·28-s − 0.557·29-s − 0.718·31-s + 0.176·32-s − 0.348·33-s − 0.171·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(6.174150462\) |
\(L(\frac12)\) |
\(\approx\) |
\(6.174150462\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 5 | \( 1 \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 + T \) |
good | 3 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - 5 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 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
−14.53047515143364, −14.05232364827576, −13.63660650095508, −13.01146632106849, −12.69856021437928, −11.90688477986616, −11.20791417589750, −11.01642054229864, −10.85956756342280, −9.903109451842595, −9.076566313571070, −8.570546083594531, −8.234736434247532, −7.802713052193847, −7.180935252697977, −6.388568266352953, −5.798057828158813, −5.292950668091946, −4.776952257535287, −3.953549233674791, −3.810315191568798, −2.632982235008043, −2.283472720558165, −1.676391106919605, −0.7369802411928645,
0.7369802411928645, 1.676391106919605, 2.283472720558165, 2.632982235008043, 3.810315191568798, 3.953549233674791, 4.776952257535287, 5.292950668091946, 5.798057828158813, 6.388568266352953, 7.180935252697977, 7.802713052193847, 8.234736434247532, 8.570546083594531, 9.076566313571070, 9.903109451842595, 10.85956756342280, 11.01642054229864, 11.20791417589750, 11.90688477986616, 12.69856021437928, 13.01146632106849, 13.63660650095508, 14.05232364827576, 14.53047515143364