| L(s) = 1 | − 13-s + 25-s + 3·31-s + 4·37-s + 3·43-s − 49-s + 61-s − 3·67-s + 2·73-s − 3·79-s − 97-s + 3·103-s + 2·109-s − 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | − 13-s + 25-s + 3·31-s + 4·37-s + 3·43-s − 49-s + 61-s − 3·67-s + 2·73-s − 3·79-s − 97-s + 3·103-s + 2·109-s − 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + 173-s + 179-s + 181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15116544 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15116544 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.668740490\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.668740490\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 7 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 31 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 37 | $C_1$ | \( ( 1 - T )^{4} \) |
| 41 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
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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
−8.885961145924870923476052039836, −8.386518690992425855669513092197, −7.992187187892519765486968507591, −7.86775293961580750167101300042, −7.32318509727496407983457496442, −7.14648041773955726321610064694, −6.59939036896005156344906845643, −6.16552541960628936440936917220, −5.90079189278106439658667161761, −5.71855337963896163041529332904, −4.77692405685990224974818861493, −4.72230224665409569264772137328, −4.34040617308450464683441648509, −4.12961318839900890955541060973, −3.07681920234615565342713484059, −3.03972133929248253682661866983, −2.38953697110489801921221712427, −2.30154171285332870199270133133, −1.04902367857313181663747407626, −0.949767843706296318295660339738,
0.949767843706296318295660339738, 1.04902367857313181663747407626, 2.30154171285332870199270133133, 2.38953697110489801921221712427, 3.03972133929248253682661866983, 3.07681920234615565342713484059, 4.12961318839900890955541060973, 4.34040617308450464683441648509, 4.72230224665409569264772137328, 4.77692405685990224974818861493, 5.71855337963896163041529332904, 5.90079189278106439658667161761, 6.16552541960628936440936917220, 6.59939036896005156344906845643, 7.14648041773955726321610064694, 7.32318509727496407983457496442, 7.86775293961580750167101300042, 7.992187187892519765486968507591, 8.386518690992425855669513092197, 8.885961145924870923476052039836
