| L(s) = 1 | + 4-s + 9-s + 12·11-s − 2·19-s − 16·31-s + 36-s + 12·41-s + 12·44-s − 13·49-s + 24·59-s − 4·61-s − 64-s − 12·71-s − 2·76-s − 20·79-s − 24·89-s + 12·99-s − 56·109-s + 58·121-s − 16·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 1/3·9-s + 3.61·11-s − 0.458·19-s − 2.87·31-s + 1/6·36-s + 1.87·41-s + 1.80·44-s − 1.85·49-s + 3.12·59-s − 0.512·61-s − 1/8·64-s − 1.42·71-s − 0.229·76-s − 2.25·79-s − 2.54·89-s + 1.20·99-s − 5.36·109-s + 5.27·121-s − 1.43·124-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6048080663\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6048080663\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 3 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 5 | | \( 1 \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| good | 7 | $C_2^2$$\times$$C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )( 1 + 11 T^{2} + p^{2} T^{4} ) \) |
| 11 | $C_2^2$ | \( ( 1 - 6 T + 25 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^3$ | \( 1 - 2 T^{2} - 285 T^{4} - 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 37 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 12 T + 107 T^{2} - 12 p T^{3} + p^{2} T^{4} )( 1 + 12 T + 107 T^{2} + 12 p T^{3} + p^{2} T^{4} ) \) |
| 41 | $C_2^2$ | \( ( 1 - 6 T - 5 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^3$ | \( 1 + 85 T^{2} + 5376 T^{4} + 85 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 12 T + 85 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 2 T - 57 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 - 35 T^{2} - 3264 T^{4} - 35 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 + 6 T - 35 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 97 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 12 T + 55 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 169 T^{2} + p^{2} T^{4} )( 1 + 167 T^{2} + p^{2} T^{4} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.71341561153483013191831073281, −6.47309418876739527727631788544, −6.02457553226833754644417273859, −5.89321852127444986574912822227, −5.70674935795786447711900676914, −5.67344122062404083669096171978, −5.43333513721371153497509025237, −4.93034327337930032598372927292, −4.77359152517997718114616625533, −4.53973180348405330736751707538, −4.24204752112278289066887257787, −4.10204012204989757804359922355, −3.88710792352370867396442538241, −3.69255811253447848907167548011, −3.49225249685605745822143179719, −3.46554564016068561104184564995, −2.78802857703759676883157314469, −2.58127202719355798520914909276, −2.45130349007716682344814051823, −2.07965330439634660882867461391, −1.57339606165677084895799091834, −1.37834878504508090906768831533, −1.32600786171351718364176670396, −1.11071445497090327956839159750, −0.11427172476290294706740214868,
0.11427172476290294706740214868, 1.11071445497090327956839159750, 1.32600786171351718364176670396, 1.37834878504508090906768831533, 1.57339606165677084895799091834, 2.07965330439634660882867461391, 2.45130349007716682344814051823, 2.58127202719355798520914909276, 2.78802857703759676883157314469, 3.46554564016068561104184564995, 3.49225249685605745822143179719, 3.69255811253447848907167548011, 3.88710792352370867396442538241, 4.10204012204989757804359922355, 4.24204752112278289066887257787, 4.53973180348405330736751707538, 4.77359152517997718114616625533, 4.93034327337930032598372927292, 5.43333513721371153497509025237, 5.67344122062404083669096171978, 5.70674935795786447711900676914, 5.89321852127444986574912822227, 6.02457553226833754644417273859, 6.47309418876739527727631788544, 6.71341561153483013191831073281
