| L(s) = 1 | − 2-s + 4-s − 2·5-s − 8-s + 9-s + 2·10-s + 2·13-s + 16-s − 18-s − 2·20-s + 3·25-s − 2·26-s − 32-s + 36-s + 4·37-s + 2·40-s + 12·41-s − 2·45-s + 2·49-s − 3·50-s + 2·52-s + 12·53-s + 4·61-s + 64-s − 4·65-s − 72-s + 16·73-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.894·5-s − 0.353·8-s + 1/3·9-s + 0.632·10-s + 0.554·13-s + 1/4·16-s − 0.235·18-s − 0.447·20-s + 3/5·25-s − 0.392·26-s − 0.176·32-s + 1/6·36-s + 0.657·37-s + 0.316·40-s + 1.87·41-s − 0.298·45-s + 2/7·49-s − 0.424·50-s + 0.277·52-s + 1.64·53-s + 0.512·61-s + 1/8·64-s − 0.496·65-s − 0.117·72-s + 1.87·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.286911463\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.286911463\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 13 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 98 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−7.993652391521439250723001751174, −7.66625670460088011141896478037, −7.23091626527372242959341239046, −6.89632635082758358357029115131, −6.30563282675575538526656888797, −5.99714563622230372484380845211, −5.36716835253816956204350340049, −4.90287820927114447450258908945, −4.18470159024800140960378103774, −3.92871883184026369104779176326, −3.41067871091058592233972496042, −2.64534656414478412609021430765, −2.22773038890382925548262920400, −1.23286094969581372261993193711, −0.64418508658265482350248534270,
0.64418508658265482350248534270, 1.23286094969581372261993193711, 2.22773038890382925548262920400, 2.64534656414478412609021430765, 3.41067871091058592233972496042, 3.92871883184026369104779176326, 4.18470159024800140960378103774, 4.90287820927114447450258908945, 5.36716835253816956204350340049, 5.99714563622230372484380845211, 6.30563282675575538526656888797, 6.89632635082758358357029115131, 7.23091626527372242959341239046, 7.66625670460088011141896478037, 7.993652391521439250723001751174
