| L(s) = 1 | − 2·7-s − 4·17-s − 6·25-s + 16·31-s − 20·41-s + 16·47-s + 3·49-s + 16·71-s − 12·73-s − 16·79-s − 20·89-s + 4·97-s + 32·103-s − 4·113-s + 8·119-s + 18·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s + 173-s + ⋯ |
| L(s) = 1 | − 0.755·7-s − 0.970·17-s − 6/5·25-s + 2.87·31-s − 3.12·41-s + 2.33·47-s + 3/7·49-s + 1.89·71-s − 1.40·73-s − 1.80·79-s − 2.11·89-s + 0.406·97-s + 3.15·103-s − 0.376·113-s + 0.733·119-s + 1.63·121-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 + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.769·169-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16257024 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16257024 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.325945232\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.325945232\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−8.594138504914472001145397232947, −8.187875093824406687881848441341, −8.113131026503448174744106376987, −7.45252346195597789964208339012, −7.02829744772771804726163696101, −6.87544041084269728372758274499, −6.34082299462303647528116045608, −6.23038406654064686657554777937, −5.53080370908755604213639003932, −5.51558701648541930492972690831, −4.71029555021160316704398631829, −4.46637698311959578082903336940, −4.15019072992526583697013643942, −3.56457307318999445538383723413, −3.18485662754625896241065754704, −2.73441637613135709485704916435, −2.26642549537357884217521702307, −1.79572639290480560241780447798, −1.06319125102406991985835289711, −0.36261959684987456762696970933,
0.36261959684987456762696970933, 1.06319125102406991985835289711, 1.79572639290480560241780447798, 2.26642549537357884217521702307, 2.73441637613135709485704916435, 3.18485662754625896241065754704, 3.56457307318999445538383723413, 4.15019072992526583697013643942, 4.46637698311959578082903336940, 4.71029555021160316704398631829, 5.51558701648541930492972690831, 5.53080370908755604213639003932, 6.23038406654064686657554777937, 6.34082299462303647528116045608, 6.87544041084269728372758274499, 7.02829744772771804726163696101, 7.45252346195597789964208339012, 8.113131026503448174744106376987, 8.187875093824406687881848441341, 8.594138504914472001145397232947
