| L(s) = 1 | − 2-s − 3·3-s + 2·4-s − 6·5-s + 3·6-s − 5·8-s + 3·9-s + 6·10-s + 3·11-s − 6·12-s + 2·13-s + 18·15-s + 5·16-s − 2·17-s − 3·18-s − 19-s − 12·20-s − 3·22-s + 15·24-s + 17·25-s − 2·26-s − 7·29-s − 18·30-s − 6·31-s − 10·32-s − 9·33-s + 2·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.73·3-s + 4-s − 2.68·5-s + 1.22·6-s − 1.76·8-s + 9-s + 1.89·10-s + 0.904·11-s − 1.73·12-s + 0.554·13-s + 4.64·15-s + 5/4·16-s − 0.485·17-s − 0.707·18-s − 0.229·19-s − 2.68·20-s − 0.639·22-s + 3.06·24-s + 17/5·25-s − 0.392·26-s − 1.29·29-s − 3.28·30-s − 1.07·31-s − 1.76·32-s − 1.56·33-s + 0.342·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405769 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405769 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 7 | | \( 1 \) |
| 13 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + T - T^{2} + p T^{3} + p^{2} T^{4} \) |
| 3 | $C_2$ | \( ( 1 + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 7 T + 20 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 3 T - 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 7 T + 6 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 4 T - 43 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 3 T - 58 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 13 T + 98 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 19 T + p T^{2} ) \) |
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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
−10.80761314769465912166222247823, −10.11119986112905739488476531203, −9.319309177067364670683997106943, −8.953065874571569495482158931145, −8.782157788270012433070236009096, −7.926414403291826510603158269533, −7.79567633373124016427401213919, −7.20892396848169732998825106118, −6.89395633091701794443273327264, −6.26385727754423665528886900574, −6.08453405277062383756315652768, −5.46005584618270116139243791811, −4.89627513052537269191172363045, −3.94114065870228253874817188542, −3.91578565359133822835240984392, −3.33453984966480936188307050410, −2.39667627224644028421078766268, −1.18585911043870389244667343960, 0, 0,
1.18585911043870389244667343960, 2.39667627224644028421078766268, 3.33453984966480936188307050410, 3.91578565359133822835240984392, 3.94114065870228253874817188542, 4.89627513052537269191172363045, 5.46005584618270116139243791811, 6.08453405277062383756315652768, 6.26385727754423665528886900574, 6.89395633091701794443273327264, 7.20892396848169732998825106118, 7.79567633373124016427401213919, 7.926414403291826510603158269533, 8.782157788270012433070236009096, 8.953065874571569495482158931145, 9.319309177067364670683997106943, 10.11119986112905739488476531203, 10.80761314769465912166222247823
