| L(s) = 1 | + 4-s − 7-s + 9·11-s + 16-s − 9·23-s + 8·25-s − 28-s + 9·29-s − 5·37-s − 11·43-s + 9·44-s − 6·49-s + 9·53-s + 64-s − 8·67-s + 18·71-s − 9·77-s + 16·79-s − 9·92-s + 8·100-s + 9·107-s − 14·109-s − 112-s + 9·116-s + 41·121-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 0.377·7-s + 2.71·11-s + 1/4·16-s − 1.87·23-s + 8/5·25-s − 0.188·28-s + 1.67·29-s − 0.821·37-s − 1.67·43-s + 1.35·44-s − 6/7·49-s + 1.23·53-s + 1/8·64-s − 0.977·67-s + 2.13·71-s − 1.02·77-s + 1.80·79-s − 0.938·92-s + 4/5·100-s + 0.870·107-s − 1.34·109-s − 0.0944·112-s + 0.835·116-s + 3.72·121-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 142884 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 142884 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.098314748\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.098314748\) |
| \(L(\frac{3}{2})\) |
|
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_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 11 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 19 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 37 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 47 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 65 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 166 T^{2} + p^{2} T^{4} \) |
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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
−9.224270313251311248695609503731, −8.926172776931655386392549760179, −8.364258383592395193438530601677, −8.008434032674441535390419881144, −7.11558059107356373378861048035, −6.74548370957534655750843248387, −6.36170574445466643227836272841, −6.21468561426115465778487864224, −5.20146507786138395849998023379, −4.68297635191979650163874975909, −3.80935334035983306252563879927, −3.68666705632161109054794719122, −2.78285416187718949509299611167, −1.84767910611996404864734320704, −1.10136425700326189141971822968,
1.10136425700326189141971822968, 1.84767910611996404864734320704, 2.78285416187718949509299611167, 3.68666705632161109054794719122, 3.80935334035983306252563879927, 4.68297635191979650163874975909, 5.20146507786138395849998023379, 6.21468561426115465778487864224, 6.36170574445466643227836272841, 6.74548370957534655750843248387, 7.11558059107356373378861048035, 8.008434032674441535390419881144, 8.364258383592395193438530601677, 8.926172776931655386392549760179, 9.224270313251311248695609503731
