| L(s) = 1 | + 4·3-s + 6·9-s − 4·17-s − 4·19-s + 6·25-s − 4·27-s − 4·41-s + 16·43-s + 49-s − 16·51-s − 16·57-s + 12·59-s − 24·67-s − 28·73-s + 24·75-s − 37·81-s + 12·83-s + 20·89-s − 4·97-s − 24·107-s + 12·113-s − 22·121-s − 16·123-s + 127-s + 64·129-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 2.30·3-s + 2·9-s − 0.970·17-s − 0.917·19-s + 6/5·25-s − 0.769·27-s − 0.624·41-s + 2.43·43-s + 1/7·49-s − 2.24·51-s − 2.11·57-s + 1.56·59-s − 2.93·67-s − 3.27·73-s + 2.77·75-s − 4.11·81-s + 1.31·83-s + 2.11·89-s − 0.406·97-s − 2.32·107-s + 1.12·113-s − 2·121-s − 1.44·123-s + 0.0887·127-s + 5.63·129-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25088 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25088 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.261235310\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.261235310\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 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
−10.76672173106036800914090760816, −9.957478933829252598803944087984, −9.271787274682453081680950548186, −8.928329697232844766585907221767, −8.676248321093563858087492291911, −8.188057214509100026281306554311, −7.51048678322729423104589438688, −7.14438434722230698367292292955, −6.28154185525062265042607550144, −5.61936194790385614498397188009, −4.50185552371175935345657331179, −4.05975063928059999821924523382, −3.14137792992390816749783057891, −2.66547510967810892791353271494, −1.94068717855646026245068222948,
1.94068717855646026245068222948, 2.66547510967810892791353271494, 3.14137792992390816749783057891, 4.05975063928059999821924523382, 4.50185552371175935345657331179, 5.61936194790385614498397188009, 6.28154185525062265042607550144, 7.14438434722230698367292292955, 7.51048678322729423104589438688, 8.188057214509100026281306554311, 8.676248321093563858087492291911, 8.928329697232844766585907221767, 9.271787274682453081680950548186, 9.957478933829252598803944087984, 10.76672173106036800914090760816
