L(s) = 1 | + 2·3-s + 3·9-s + 8·11-s − 12·17-s − 8·19-s − 6·25-s + 4·27-s + 16·33-s + 4·41-s + 8·43-s + 2·49-s − 24·51-s − 16·57-s − 8·59-s + 8·67-s − 12·73-s − 12·75-s + 5·81-s + 24·83-s + 20·89-s − 28·97-s + 24·99-s − 8·107-s + 4·113-s + 26·121-s + 8·123-s + 127-s + ⋯ |
L(s) = 1 | + 1.15·3-s + 9-s + 2.41·11-s − 2.91·17-s − 1.83·19-s − 6/5·25-s + 0.769·27-s + 2.78·33-s + 0.624·41-s + 1.21·43-s + 2/7·49-s − 3.36·51-s − 2.11·57-s − 1.04·59-s + 0.977·67-s − 1.40·73-s − 1.38·75-s + 5/9·81-s + 2.63·83-s + 2.11·89-s − 2.84·97-s + 2.41·99-s − 0.773·107-s + 0.376·113-s + 2.36·121-s + 0.721·123-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18432 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18432 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.658422999\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.658422999\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
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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
−11.01151255843213055292894337270, −10.44913304188171401892835914004, −9.488699237661810053920246470193, −9.240123327709471504547415939693, −8.839886292016704738441339481672, −8.460175078145674768211752918036, −7.67780945046588747720337476950, −6.91469103626909406223255043804, −6.44896944953500655110243120870, −6.15267013015603292112906972931, −4.62570703678849832502580328887, −4.10625798014363516984369255368, −3.85566973953101374011358603281, −2.44821729992400328434125228202, −1.83689405023894510739841616557,
1.83689405023894510739841616557, 2.44821729992400328434125228202, 3.85566973953101374011358603281, 4.10625798014363516984369255368, 4.62570703678849832502580328887, 6.15267013015603292112906972931, 6.44896944953500655110243120870, 6.91469103626909406223255043804, 7.67780945046588747720337476950, 8.460175078145674768211752918036, 8.839886292016704738441339481672, 9.240123327709471504547415939693, 9.488699237661810053920246470193, 10.44913304188171401892835914004, 11.01151255843213055292894337270