| L(s) = 1 | + 3-s − 4-s + 9-s − 12-s + 2·13-s − 3·16-s − 10·25-s + 27-s − 36-s + 2·39-s − 8·43-s − 3·48-s + 2·49-s − 2·52-s + 4·61-s + 7·64-s − 10·75-s + 16·79-s + 81-s + 10·100-s + 16·103-s − 108-s + 2·117-s − 10·121-s + 127-s − 8·129-s + 131-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1/2·4-s + 1/3·9-s − 0.288·12-s + 0.554·13-s − 3/4·16-s − 2·25-s + 0.192·27-s − 1/6·36-s + 0.320·39-s − 1.21·43-s − 0.433·48-s + 2/7·49-s − 0.277·52-s + 0.512·61-s + 7/8·64-s − 1.15·75-s + 1.80·79-s + 1/9·81-s + 100-s + 1.57·103-s − 0.0962·108-s + 0.184·117-s − 0.909·121-s + 0.0887·127-s − 0.704·129-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4563 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4563 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8688739290\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8688739290\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( 1 - T \) |
| 13 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 82 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 130 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 154 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 130 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−12.37022875262449636944115326377, −11.75021766055426081014441395158, −11.26214871088196986969621364833, −10.50362814369017161678338427353, −9.822547030359767529196509389130, −9.394110857034161108190630023282, −8.716178163971269186033685676944, −8.187549007456376769469535656463, −7.54986009243878845014659485386, −6.72746551951994728812600570829, −5.98311934439363478319193558532, −5.07052661557595651641559047460, −4.18486743734375640931684483257, −3.47002734719492040287822111515, −2.09536210327137556917765822110,
2.09536210327137556917765822110, 3.47002734719492040287822111515, 4.18486743734375640931684483257, 5.07052661557595651641559047460, 5.98311934439363478319193558532, 6.72746551951994728812600570829, 7.54986009243878845014659485386, 8.187549007456376769469535656463, 8.716178163971269186033685676944, 9.394110857034161108190630023282, 9.822547030359767529196509389130, 10.50362814369017161678338427353, 11.26214871088196986969621364833, 11.75021766055426081014441395158, 12.37022875262449636944115326377
