L(s) = 1 | − 2-s + 4-s − 2·8-s − 9-s + 11-s + 2·16-s + 18-s − 22-s − 23-s − 2·29-s − 2·32-s − 36-s − 37-s + 2·43-s + 44-s + 46-s + 2·53-s + 2·58-s + 3·64-s − 67-s − 2·71-s + 2·72-s + 74-s + 79-s − 2·86-s − 2·88-s − 92-s + ⋯ |
L(s) = 1 | − 2-s + 4-s − 2·8-s − 9-s + 11-s + 2·16-s + 18-s − 22-s − 23-s − 2·29-s − 2·32-s − 36-s − 37-s + 2·43-s + 44-s + 46-s + 2·53-s + 2·58-s + 3·64-s − 67-s − 2·71-s + 2·72-s + 74-s + 79-s − 2·86-s − 2·88-s − 92-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1500625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1500625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4857522993\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4857522993\) |
\(L(1)\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
good | 2 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 3 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
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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
−10.03313300817045812074929488173, −9.578040823003165920550467263826, −9.229758580470922955291942117830, −8.924255916202679686666584223595, −8.576136475883728372990583116844, −8.372531608678366215526791706188, −7.59842176902218999053938089307, −7.30725607402941585738144896994, −7.04006346212564303124405887976, −6.19099761555383735231315200215, −6.13763063646913018186982607990, −5.57428619416559246494646842947, −5.49525113082812853994718989058, −4.37372423176520755756869262364, −3.98645908157874496953427827379, −3.20266447089468753644929897930, −3.15985079605847204754569431114, −2.10925468469655311819240858735, −1.95256991266311545215048999760, −0.69949591763161655647229985242,
0.69949591763161655647229985242, 1.95256991266311545215048999760, 2.10925468469655311819240858735, 3.15985079605847204754569431114, 3.20266447089468753644929897930, 3.98645908157874496953427827379, 4.37372423176520755756869262364, 5.49525113082812853994718989058, 5.57428619416559246494646842947, 6.13763063646913018186982607990, 6.19099761555383735231315200215, 7.04006346212564303124405887976, 7.30725607402941585738144896994, 7.59842176902218999053938089307, 8.372531608678366215526791706188, 8.576136475883728372990583116844, 8.924255916202679686666584223595, 9.229758580470922955291942117830, 9.578040823003165920550467263826, 10.03313300817045812074929488173