| L(s) = 1 | + 2·2-s − 4-s − 10·7-s − 8·8-s + 2·9-s + 4·13-s − 20·14-s − 7·16-s + 4·18-s + 8·26-s + 10·28-s + 2·29-s + 14·32-s − 2·36-s + 8·37-s − 14·47-s + 61·49-s − 4·52-s + 80·56-s + 4·58-s + 2·61-s − 20·63-s + 35·64-s + 6·67-s − 16·72-s − 8·73-s + 16·74-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1/2·4-s − 3.77·7-s − 2.82·8-s + 2/3·9-s + 1.10·13-s − 5.34·14-s − 7/4·16-s + 0.942·18-s + 1.56·26-s + 1.88·28-s + 0.371·29-s + 2.47·32-s − 1/3·36-s + 1.31·37-s − 2.04·47-s + 61/7·49-s − 0.554·52-s + 10.6·56-s + 0.525·58-s + 0.256·61-s − 2.51·63-s + 35/8·64-s + 0.733·67-s − 1.88·72-s − 0.936·73-s + 1.85·74-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9128531535\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9128531535\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 13 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 9 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 61 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 97 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 109 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 146 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.31017665064930935619432735106, −11.69444641333483352070915432507, −10.94359805970180455673704250792, −10.18292889967561324323116744434, −9.824905212786396515507962962580, −9.551136668652097020734700477737, −9.348761342303144871886242766848, −8.621712734471143870661999535949, −8.315337011117911518015507033346, −7.12726850467944314828658979872, −6.76102674273666522072594064665, −6.18477877716931107907457986216, −6.03883018900617025019053152179, −5.55501313256729649864546186147, −4.62149057130139708810611235015, −4.01648289809901815639572885461, −3.70965379229977940462542516538, −3.01544079190678691384995975550, −2.94646835826188193449742398189, −0.54569473341836565898502453766,
0.54569473341836565898502453766, 2.94646835826188193449742398189, 3.01544079190678691384995975550, 3.70965379229977940462542516538, 4.01648289809901815639572885461, 4.62149057130139708810611235015, 5.55501313256729649864546186147, 6.03883018900617025019053152179, 6.18477877716931107907457986216, 6.76102674273666522072594064665, 7.12726850467944314828658979872, 8.315337011117911518015507033346, 8.621712734471143870661999535949, 9.348761342303144871886242766848, 9.551136668652097020734700477737, 9.824905212786396515507962962580, 10.18292889967561324323116744434, 10.94359805970180455673704250792, 11.69444641333483352070915432507, 12.31017665064930935619432735106
