| L(s) = 1 | − 2·2-s + 3-s + 2·4-s − 2·6-s + 9-s − 5·11-s + 2·12-s + 13-s − 4·16-s + 5·17-s − 2·18-s − 2·19-s + 10·22-s + 23-s − 2·26-s + 27-s + 10·29-s + 2·31-s + 8·32-s − 5·33-s − 10·34-s + 2·36-s + 3·37-s + 4·38-s + 39-s + 9·41-s + 4·43-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 0.577·3-s + 4-s − 0.816·6-s + 1/3·9-s − 1.50·11-s + 0.577·12-s + 0.277·13-s − 16-s + 1.21·17-s − 0.471·18-s − 0.458·19-s + 2.13·22-s + 0.208·23-s − 0.392·26-s + 0.192·27-s + 1.85·29-s + 0.359·31-s + 1.41·32-s − 0.870·33-s − 1.71·34-s + 1/3·36-s + 0.493·37-s + 0.648·38-s + 0.160·39-s + 1.40·41-s + 0.609·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.586913483\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.586913483\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 + p T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 11 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 15 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 11 T + p T^{2} \) |
| 97 | \( 1 + 9 T + p T^{2} \) |
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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
−14.43913844645003, −14.17984118547039, −13.62011807388121, −12.92169388650719, −12.61621622985925, −11.94961917263172, −11.15931454077530, −10.72571358492815, −10.28812877646407, −9.864071112448425, −9.368623699701371, −8.709545231166088, −8.285143286113601, −7.779113903433311, −7.580033103856334, −6.774378158432202, −6.177997012163910, −5.401962248344214, −4.771517578291045, −4.129531595549746, −3.239152322949533, −2.547783530588825, −2.188791230472139, −1.047765564949570, −0.6655789494363450,
0.6655789494363450, 1.047765564949570, 2.188791230472139, 2.547783530588825, 3.239152322949533, 4.129531595549746, 4.771517578291045, 5.401962248344214, 6.177997012163910, 6.774378158432202, 7.580033103856334, 7.779113903433311, 8.285143286113601, 8.709545231166088, 9.368623699701371, 9.864071112448425, 10.28812877646407, 10.72571358492815, 11.15931454077530, 11.94961917263172, 12.61621622985925, 12.92169388650719, 13.62011807388121, 14.17984118547039, 14.43913844645003
