| L(s) = 1 | − 2-s + 3·3-s − 3·6-s + 8-s + 6·9-s + 3·11-s + 2·13-s − 16-s + 6·17-s − 6·18-s + 2·19-s − 3·22-s + 6·23-s + 3·24-s + 5·25-s − 2·26-s + 9·27-s − 6·29-s − 4·31-s + 9·33-s − 6·34-s − 8·37-s − 2·38-s + 6·39-s + 9·41-s + 43-s − 6·46-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.73·3-s − 1.22·6-s + 0.353·8-s + 2·9-s + 0.904·11-s + 0.554·13-s − 1/4·16-s + 1.45·17-s − 1.41·18-s + 0.458·19-s − 0.639·22-s + 1.25·23-s + 0.612·24-s + 25-s − 0.392·26-s + 1.73·27-s − 1.11·29-s − 0.718·31-s + 1.56·33-s − 1.02·34-s − 1.31·37-s − 0.324·38-s + 0.960·39-s + 1.40·41-s + 0.152·43-s − 0.884·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.361626778\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.361626778\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 3 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 6 T + 7 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 9 T + 40 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 6 T - 11 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 3 T - 50 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 8 T + 3 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 12 T + 61 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 19 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−10.08638922281481078873384580728, −9.903032274368527798652125197368, −9.210979260704216312337666622340, −9.040276262268418094105582611928, −8.789105179004258680002390756483, −8.487544419431382483186701733064, −7.85634586241254813168047375829, −7.44059151717152319048962369091, −7.13658052134032557412985686404, −6.92928032823744119478600769676, −6.02555638910057487605016428280, −5.53762830792010891679050044600, −5.07335911007181134607128627244, −4.19667755000963154648833107927, −3.96481883176869853734896227193, −3.29962880189108273672831164416, −3.03926261833586846542103257069, −2.25798378495195265722827178245, −1.42891684345548479434657585203, −1.08028594500863224715436795326,
1.08028594500863224715436795326, 1.42891684345548479434657585203, 2.25798378495195265722827178245, 3.03926261833586846542103257069, 3.29962880189108273672831164416, 3.96481883176869853734896227193, 4.19667755000963154648833107927, 5.07335911007181134607128627244, 5.53762830792010891679050044600, 6.02555638910057487605016428280, 6.92928032823744119478600769676, 7.13658052134032557412985686404, 7.44059151717152319048962369091, 7.85634586241254813168047375829, 8.487544419431382483186701733064, 8.789105179004258680002390756483, 9.040276262268418094105582611928, 9.210979260704216312337666622340, 9.903032274368527798652125197368, 10.08638922281481078873384580728
