| L(s) = 1 | + 2·2-s − 2·3-s − 4-s − 4·6-s + 6·7-s − 8·8-s + 2·9-s + 2·12-s − 4·13-s + 12·14-s − 7·16-s + 4·18-s + 6·19-s − 12·21-s + 16·23-s + 16·24-s − 8·26-s − 6·27-s − 6·28-s − 14·29-s + 6·31-s + 14·32-s − 2·36-s + 2·37-s + 12·38-s + 8·39-s − 24·42-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1.15·3-s − 1/2·4-s − 1.63·6-s + 2.26·7-s − 2.82·8-s + 2/3·9-s + 0.577·12-s − 1.10·13-s + 3.20·14-s − 7/4·16-s + 0.942·18-s + 1.37·19-s − 2.61·21-s + 3.33·23-s + 3.26·24-s − 1.56·26-s − 1.15·27-s − 1.13·28-s − 2.59·29-s + 1.07·31-s + 2.47·32-s − 1/3·36-s + 0.328·37-s + 1.94·38-s + 1.28·39-s − 3.70·42-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 855625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.008414201\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.008414201\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 37 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 3 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−10.36064411301665304608862811310, −9.716834704066526040423319756452, −9.585596485488115110706128554659, −9.092096962415640318575093717439, −8.562738078495050675934083390182, −8.198988079779502516560133945599, −7.72421514620521137727135521916, −7.17555806445055832104720949487, −6.82110386031872740768304699598, −6.16533800907379519059950500906, −5.43443841568821475809739782313, −5.19772248993176353649746483620, −5.03574541911461831534542273599, −4.93441539089070830944985017230, −4.33800225605344126432721724490, −3.51225529781775833170540860983, −3.34474249918042278093347192068, −2.35822040831558650167755421969, −1.46757420986876449562073027714, −0.62730601236028749827165058491,
0.62730601236028749827165058491, 1.46757420986876449562073027714, 2.35822040831558650167755421969, 3.34474249918042278093347192068, 3.51225529781775833170540860983, 4.33800225605344126432721724490, 4.93441539089070830944985017230, 5.03574541911461831534542273599, 5.19772248993176353649746483620, 5.43443841568821475809739782313, 6.16533800907379519059950500906, 6.82110386031872740768304699598, 7.17555806445055832104720949487, 7.72421514620521137727135521916, 8.198988079779502516560133945599, 8.562738078495050675934083390182, 9.092096962415640318575093717439, 9.585596485488115110706128554659, 9.716834704066526040423319756452, 10.36064411301665304608862811310
