| L(s) = 1 | − 2-s − 2·3-s + 2·4-s + 2·5-s + 2·6-s − 5·8-s + 3·9-s − 2·10-s − 11-s − 4·12-s + 8·13-s − 4·15-s + 5·16-s − 4·17-s − 3·18-s + 4·20-s + 22-s + 4·23-s + 10·24-s + 5·25-s − 8·26-s − 10·27-s − 12·29-s + 4·30-s − 10·31-s − 10·32-s + 2·33-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s + 4-s + 0.894·5-s + 0.816·6-s − 1.76·8-s + 9-s − 0.632·10-s − 0.301·11-s − 1.15·12-s + 2.21·13-s − 1.03·15-s + 5/4·16-s − 0.970·17-s − 0.707·18-s + 0.894·20-s + 0.213·22-s + 0.834·23-s + 2.04·24-s + 25-s − 1.56·26-s − 1.92·27-s − 2.22·29-s + 0.730·30-s − 1.79·31-s − 1.76·32-s + 0.348·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 290521 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 290521 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.004437336\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.004437336\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | | \( 1 \) |
| 11 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + T - T^{2} + p T^{3} + p^{2} T^{4} \) |
| 3 | $C_2^2$ | \( 1 + 2 T + T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 4 T - T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 4 T - 7 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 10 T + 69 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 6 T - T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 10 T + 53 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 2 T - 55 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 8 T - 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 8 T - 9 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−10.97648437404363202733622829791, −10.94631785250250540258437311197, −10.36149645196693557360741551364, −9.543965587989615812576794801884, −9.255607369495426077510239146831, −8.915068724933190951973655638418, −8.787498650057598666525220333178, −7.71430761165178166621451971517, −7.24874020720812793931931323741, −7.14289908071580893222445482721, −6.12008228585343471143273714559, −6.07904090595396138305132622536, −5.61367086313176459278527558449, −5.58613287538501873009718315485, −4.21974800368305056021870637145, −3.95104844302004535037014515030, −3.04092962074346211991971874253, −2.29747422209458246179664832279, −1.65851478706095746306995211881, −0.71514091490976460271343380200,
0.71514091490976460271343380200, 1.65851478706095746306995211881, 2.29747422209458246179664832279, 3.04092962074346211991971874253, 3.95104844302004535037014515030, 4.21974800368305056021870637145, 5.58613287538501873009718315485, 5.61367086313176459278527558449, 6.07904090595396138305132622536, 6.12008228585343471143273714559, 7.14289908071580893222445482721, 7.24874020720812793931931323741, 7.71430761165178166621451971517, 8.787498650057598666525220333178, 8.915068724933190951973655638418, 9.255607369495426077510239146831, 9.543965587989615812576794801884, 10.36149645196693557360741551364, 10.94631785250250540258437311197, 10.97648437404363202733622829791
