| L(s) = 1 | − 2·2-s + 2·4-s − 4·5-s + 6·7-s − 9-s + 8·10-s + 2·11-s + 8·13-s − 12·14-s − 4·16-s + 6·17-s + 2·18-s − 6·19-s − 8·20-s − 4·22-s − 2·23-s + 11·25-s − 16·26-s + 12·28-s + 6·29-s + 8·32-s − 12·34-s − 24·35-s − 2·36-s + 16·37-s + 12·38-s − 12·43-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s − 1.78·5-s + 2.26·7-s − 1/3·9-s + 2.52·10-s + 0.603·11-s + 2.21·13-s − 3.20·14-s − 16-s + 1.45·17-s + 0.471·18-s − 1.37·19-s − 1.78·20-s − 0.852·22-s − 0.417·23-s + 11/5·25-s − 3.13·26-s + 2.26·28-s + 1.11·29-s + 1.41·32-s − 2.05·34-s − 4.05·35-s − 1/3·36-s + 2.63·37-s + 1.94·38-s − 1.82·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 57600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7393776148\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7393776148\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 18 T + 162 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
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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
−11.96024738045319848409027760090, −11.67709057056776640608665440554, −11.25747202673573245353186410792, −11.05892076509857134572949158698, −10.48926098466320785324298423132, −10.24645145614500882936019503435, −9.022944640304182734043171508817, −8.847102673013975322189811811695, −8.365898651857278957392919164717, −8.106713653208418927705469847711, −7.58987157855732880100126859742, −7.50217387271728220135325116551, −6.23036938681267556806264616869, −6.12505135261452938585231677078, −4.78240645545692724880683043666, −4.46951425297280730473998307454, −3.94419874086849339702377654713, −3.03414532889574315065370727459, −1.59657543727113414694446654062, −1.05186215761897664409518131513,
1.05186215761897664409518131513, 1.59657543727113414694446654062, 3.03414532889574315065370727459, 3.94419874086849339702377654713, 4.46951425297280730473998307454, 4.78240645545692724880683043666, 6.12505135261452938585231677078, 6.23036938681267556806264616869, 7.50217387271728220135325116551, 7.58987157855732880100126859742, 8.106713653208418927705469847711, 8.365898651857278957392919164717, 8.847102673013975322189811811695, 9.022944640304182734043171508817, 10.24645145614500882936019503435, 10.48926098466320785324298423132, 11.05892076509857134572949158698, 11.25747202673573245353186410792, 11.67709057056776640608665440554, 11.96024738045319848409027760090
