L(s) = 1 | + 2-s − 3-s − 4-s − 5-s − 6-s − 3·8-s + 9-s − 10-s + 8·11-s + 12-s + 15-s − 16-s − 4·17-s + 18-s + 20-s + 8·22-s + 3·24-s + 25-s − 27-s + 30-s + 5·32-s − 8·33-s − 4·34-s − 36-s + 3·40-s + 8·43-s − 8·44-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.447·5-s − 0.408·6-s − 1.06·8-s + 1/3·9-s − 0.316·10-s + 2.41·11-s + 0.288·12-s + 0.258·15-s − 1/4·16-s − 0.970·17-s + 0.235·18-s + 0.223·20-s + 1.70·22-s + 0.612·24-s + 1/5·25-s − 0.192·27-s + 0.182·30-s + 0.883·32-s − 1.39·33-s − 0.685·34-s − 1/6·36-s + 0.474·40-s + 1.21·43-s − 1.20·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.315775981\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.315775981\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 2 | $C_2$ | \( 1 - T + p T^{2} \) |
| 3 | $C_1$ | \( 1 + T \) |
| 5 | $C_1$ | \( 1 + T \) |
good | 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−9.738851148469479131038939478590, −9.655893575783761174602693945437, −9.049118048765622993735733025092, −8.554588868470699894187777077269, −8.151425844540532890281788534319, −7.10478788988449333222322211892, −6.76955885158912340897756950945, −6.34199635143511529870167101224, −5.72078508715635747921690343280, −5.04860090093668311608273372623, −4.42801570520906772580093162574, −3.86223113294681320091712334301, −3.63412818236731287070311550143, −2.32158421366848578869138486097, −0.957897388351054006923114027189,
0.957897388351054006923114027189, 2.32158421366848578869138486097, 3.63412818236731287070311550143, 3.86223113294681320091712334301, 4.42801570520906772580093162574, 5.04860090093668311608273372623, 5.72078508715635747921690343280, 6.34199635143511529870167101224, 6.76955885158912340897756950945, 7.10478788988449333222322211892, 8.151425844540532890281788534319, 8.554588868470699894187777077269, 9.049118048765622993735733025092, 9.655893575783761174602693945437, 9.738851148469479131038939478590