| L(s) = 1 | − 3·5-s − 6·7-s + 3·9-s − 2·11-s − 2·13-s + 2·17-s + 19-s − 8·23-s + 5·25-s − 20·31-s + 18·35-s − 6·37-s + 4·41-s − 4·43-s − 9·45-s + 6·47-s + 13·49-s − 5·53-s + 6·55-s − 4·59-s + 12·61-s − 18·63-s + 6·65-s − 12·67-s − 2·71-s − 6·73-s + 12·77-s + ⋯ |
| L(s) = 1 | − 1.34·5-s − 2.26·7-s + 9-s − 0.603·11-s − 0.554·13-s + 0.485·17-s + 0.229·19-s − 1.66·23-s + 25-s − 3.59·31-s + 3.04·35-s − 0.986·37-s + 0.624·41-s − 0.609·43-s − 1.34·45-s + 0.875·47-s + 13/7·49-s − 0.686·53-s + 0.809·55-s − 0.520·59-s + 1.53·61-s − 2.26·63-s + 0.744·65-s − 1.46·67-s − 0.237·71-s − 0.702·73-s + 1.36·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 698896 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 698896 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| 19 | $C_2$ | \( 1 - T + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $C_2^2$ | \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 8 T + 41 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 4 T - 25 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 4 T - 27 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 6 T - 11 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 5 T - 28 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 4 T - 43 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 12 T + 83 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 12 T + 77 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 2 T - 67 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 6 T - 37 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 5 T - 54 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 19 T + p T^{2} ) \) |
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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
−10.10754373003765670065303034661, −9.553368263832911304227745099022, −9.123330861885296845841811709405, −9.040364398310377675422377231355, −7.989697230363841404693606409503, −7.926826907199883880787553631351, −7.23322335428863357566080593188, −7.20911491917761151247348398887, −6.64713277573709279939331199353, −6.15723598588027672343295172100, −5.49297859252201295549269842583, −5.27470741613562768127774106500, −4.30016703800250525708954826112, −3.97562046723761997317968570866, −3.52445257000607845336225941310, −3.21002949373288489269920743629, −2.44609052511769958182829632071, −1.58210921019814755120221003771, 0, 0,
1.58210921019814755120221003771, 2.44609052511769958182829632071, 3.21002949373288489269920743629, 3.52445257000607845336225941310, 3.97562046723761997317968570866, 4.30016703800250525708954826112, 5.27470741613562768127774106500, 5.49297859252201295549269842583, 6.15723598588027672343295172100, 6.64713277573709279939331199353, 7.20911491917761151247348398887, 7.23322335428863357566080593188, 7.926826907199883880787553631351, 7.989697230363841404693606409503, 9.040364398310377675422377231355, 9.123330861885296845841811709405, 9.553368263832911304227745099022, 10.10754373003765670065303034661
