| L(s) = 1 | + 4-s + 7-s + 3·19-s + 28-s − 37-s − 4·43-s − 3·61-s − 64-s + 67-s + 3·73-s + 3·76-s + 79-s + 3·103-s − 109-s + 121-s + 127-s + 131-s + 3·133-s + 137-s + 139-s − 148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s + ⋯ |
| L(s) = 1 | + 4-s + 7-s + 3·19-s + 28-s − 37-s − 4·43-s − 3·61-s − 64-s + 67-s + 3·73-s + 3·76-s + 79-s + 3·103-s − 109-s + 121-s + 127-s + 131-s + 3·133-s + 137-s + 139-s − 148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2480625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2480625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.749818988\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.749818988\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 11 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 29 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 43 | $C_1$ | \( ( 1 + T )^{4} \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
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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
−9.759633078010087114845882513777, −9.565360558033900248364380465734, −8.909757361405363848103438008072, −8.715781220519617269067992459067, −8.010216691182826770194000175682, −7.68824493082618607930019820666, −7.65024385460216583404467428725, −6.99329281199656667288320251889, −6.51519360316686909037643447562, −6.47562218671681984953798436746, −5.60313709408012586111736879051, −5.25775626116668737001177135895, −4.91471620087078538433950296498, −4.66840421544948942162143250317, −3.56203413978644367886573327063, −3.43807187385475128884232854567, −2.96215741090627392784972702705, −2.16877986192885348746052775847, −1.68604251927303734649373186138, −1.20230275749537431270299909491,
1.20230275749537431270299909491, 1.68604251927303734649373186138, 2.16877986192885348746052775847, 2.96215741090627392784972702705, 3.43807187385475128884232854567, 3.56203413978644367886573327063, 4.66840421544948942162143250317, 4.91471620087078538433950296498, 5.25775626116668737001177135895, 5.60313709408012586111736879051, 6.47562218671681984953798436746, 6.51519360316686909037643447562, 6.99329281199656667288320251889, 7.65024385460216583404467428725, 7.68824493082618607930019820666, 8.010216691182826770194000175682, 8.715781220519617269067992459067, 8.909757361405363848103438008072, 9.565360558033900248364380465734, 9.759633078010087114845882513777
