| L(s) = 1 | + 2·9-s + 2·13-s + 4·17-s − 5·25-s + 4·29-s + 8·37-s + 49-s − 4·53-s + 2·61-s + 8·73-s − 5·81-s − 4·89-s − 20·97-s + 18·101-s + 36·113-s + 4·117-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 8·153-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 2/3·9-s + 0.554·13-s + 0.970·17-s − 25-s + 0.742·29-s + 1.31·37-s + 1/7·49-s − 0.549·53-s + 0.256·61-s + 0.936·73-s − 5/9·81-s − 0.423·89-s − 2.03·97-s + 1.79·101-s + 3.38·113-s + 0.369·117-s + 2/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.646·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 156800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.825449404\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.825449404\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + p T^{2} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 42 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 78 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 98 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 42 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 126 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
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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
−9.419124773708891129308207932368, −8.637451961851626896916670091139, −8.357173854981064477769174493101, −7.73882385996796925746037698793, −7.42102242821590724428126245649, −6.84889707781057677609336373180, −6.14994821983276956948371234409, −5.93727749758256137313305255440, −5.20351189798132700767454931912, −4.60803660767118359038432082748, −4.04098913799313794363952327959, −3.48239552274150474931307849006, −2.76385560663519897414114158856, −1.86604900247770752687661751869, −0.976324474133299963291105957675,
0.976324474133299963291105957675, 1.86604900247770752687661751869, 2.76385560663519897414114158856, 3.48239552274150474931307849006, 4.04098913799313794363952327959, 4.60803660767118359038432082748, 5.20351189798132700767454931912, 5.93727749758256137313305255440, 6.14994821983276956948371234409, 6.84889707781057677609336373180, 7.42102242821590724428126245649, 7.73882385996796925746037698793, 8.357173854981064477769174493101, 8.637451961851626896916670091139, 9.419124773708891129308207932368
