| L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s − 2·7-s + 8-s + 9-s + 10-s + 2·11-s + 12-s + 4·13-s − 2·14-s + 15-s + 16-s − 2·17-s + 18-s − 19-s + 20-s − 2·21-s + 2·22-s + 4·23-s + 24-s + 25-s + 4·26-s + 27-s − 2·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 0.755·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.603·11-s + 0.288·12-s + 1.10·13-s − 0.534·14-s + 0.258·15-s + 1/4·16-s − 0.485·17-s + 0.235·18-s − 0.229·19-s + 0.223·20-s − 0.436·21-s + 0.426·22-s + 0.834·23-s + 0.204·24-s + 1/5·25-s + 0.784·26-s + 0.192·27-s − 0.377·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.800031251\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.800031251\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−10.85725384968933766949121476212, −9.771329291313962369487863150151, −9.067929870365105743311066246139, −8.120082578666039881175502251759, −6.78005489199886281541883353311, −6.34035147897517033626324861495, −5.11276086658087810706615114559, −3.86805091123801542174694378598, −3.08594508663852043388720199555, −1.68107245455287324821730445547,
1.68107245455287324821730445547, 3.08594508663852043388720199555, 3.86805091123801542174694378598, 5.11276086658087810706615114559, 6.34035147897517033626324861495, 6.78005489199886281541883353311, 8.120082578666039881175502251759, 9.067929870365105743311066246139, 9.771329291313962369487863150151, 10.85725384968933766949121476212
