| L(s) = 1 | − 2-s + 3-s − 6-s − 2·7-s + 8-s + 3·9-s − 12·11-s + 5·13-s + 2·14-s − 16-s − 3·17-s − 3·18-s − 2·21-s + 12·22-s − 3·23-s + 24-s + 5·25-s − 5·26-s + 8·27-s + 9·29-s + 8·31-s − 12·33-s + 3·34-s − 4·37-s + 5·39-s + 2·42-s − 8·43-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s − 0.408·6-s − 0.755·7-s + 0.353·8-s + 9-s − 3.61·11-s + 1.38·13-s + 0.534·14-s − 1/4·16-s − 0.727·17-s − 0.707·18-s − 0.436·21-s + 2.55·22-s − 0.625·23-s + 0.204·24-s + 25-s − 0.980·26-s + 1.53·27-s + 1.67·29-s + 1.43·31-s − 2.08·33-s + 0.514·34-s − 0.657·37-s + 0.800·39-s + 0.308·42-s − 1.21·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 521284 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 521284 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9134611731\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9134611731\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 19 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 3 T - 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 9 T + 52 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 3 T - 44 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 9 T + 22 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 10 T + 39 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 6 T - 35 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 10 T + 21 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 12 T + 55 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 10 T + 3 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
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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
−10.53077602142962907183313860133, −10.22101133251579735860213114071, −9.796749249224507096433177692171, −9.546760410176431520553977692956, −8.537926150676000485622522402607, −8.419819136123545691116628041944, −8.239521637892575084687539446303, −7.942568995500605341179616020238, −7.11712316196877396218758258705, −6.69230721480135968216691032129, −6.57537934232554710442037647549, −5.50040037976405017813878689585, −5.36661437045520252495305862638, −4.47195322090585336419526387351, −4.45902863949864490936020962716, −3.28887938852658383790051091168, −2.88917259317035582843155482657, −2.56942305234563306728464640756, −1.61558824482515643800911368613, −0.54032630809861117031678362844,
0.54032630809861117031678362844, 1.61558824482515643800911368613, 2.56942305234563306728464640756, 2.88917259317035582843155482657, 3.28887938852658383790051091168, 4.45902863949864490936020962716, 4.47195322090585336419526387351, 5.36661437045520252495305862638, 5.50040037976405017813878689585, 6.57537934232554710442037647549, 6.69230721480135968216691032129, 7.11712316196877396218758258705, 7.942568995500605341179616020238, 8.239521637892575084687539446303, 8.419819136123545691116628041944, 8.537926150676000485622522402607, 9.546760410176431520553977692956, 9.796749249224507096433177692171, 10.22101133251579735860213114071, 10.53077602142962907183313860133
