| L(s) = 1 | − 4·4-s − 20·13-s + 12·16-s − 8·19-s − 4·31-s + 16·37-s + 4·43-s + 10·49-s + 80·52-s − 8·61-s − 32·64-s − 20·67-s + 32·76-s + 56·79-s + 20·97-s − 32·109-s + 16·124-s + 127-s + 131-s + 137-s + 139-s − 64·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | − 2·4-s − 5.54·13-s + 3·16-s − 1.83·19-s − 0.718·31-s + 2.63·37-s + 0.609·43-s + 10/7·49-s + 11.0·52-s − 1.02·61-s − 4·64-s − 2.44·67-s + 3.67·76-s + 6.30·79-s + 2.03·97-s − 3.06·109-s + 1.43·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 5.26·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3474305567\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3474305567\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^3$ | \( 1 + 31 T^{4} + p^{4} T^{8} \) |
| 7 | $C_2^2$ | \( ( 1 - 5 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^3$ | \( 1 + 199 T^{4} + p^{4} T^{8} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 16 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^3$ | \( 1 - 1646 T^{4} + p^{4} T^{8} \) |
| 31 | $C_2$ | \( ( 1 + T + p T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 74 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 76 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^3$ | \( 1 - 5393 T^{4} + p^{4} T^{8} \) |
| 59 | $C_2^3$ | \( 1 - 6638 T^{4} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 140 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 137 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{4} \) |
| 83 | $C_2^3$ | \( 1 - 89 T^{4} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 + 64 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{4} \) |
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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
−8.038518637428708863930756060572, −7.74298213393965585962182826168, −7.73842040573584713080577347079, −7.44723686624247340613608639382, −7.30927063210255384551126033903, −6.94367720843916167097916232154, −6.50356323943057660843832123400, −6.34585791945191617557791159134, −6.12624928219564939784020638227, −5.55988523313041032286622566118, −5.32756047011352069384440388161, −5.28219412154020868924249274055, −4.96476869240275290623446340715, −4.58124967098889192171986960913, −4.48897787248475370359130872160, −4.24395745802211909659566580048, −4.23884246784672676153777627001, −3.51804922440365647904792062450, −3.20839331520026772562491140943, −2.78527710980162340662330233998, −2.37717359825202617195487732572, −2.27985285564345268321863126614, −1.89194285751952004614765969622, −0.74746319509705047448031245802, −0.31730427996072659293550374712,
0.31730427996072659293550374712, 0.74746319509705047448031245802, 1.89194285751952004614765969622, 2.27985285564345268321863126614, 2.37717359825202617195487732572, 2.78527710980162340662330233998, 3.20839331520026772562491140943, 3.51804922440365647904792062450, 4.23884246784672676153777627001, 4.24395745802211909659566580048, 4.48897787248475370359130872160, 4.58124967098889192171986960913, 4.96476869240275290623446340715, 5.28219412154020868924249274055, 5.32756047011352069384440388161, 5.55988523313041032286622566118, 6.12624928219564939784020638227, 6.34585791945191617557791159134, 6.50356323943057660843832123400, 6.94367720843916167097916232154, 7.30927063210255384551126033903, 7.44723686624247340613608639382, 7.73842040573584713080577347079, 7.74298213393965585962182826168, 8.038518637428708863930756060572
