| L(s) = 1 | − 39·5-s − 70·7-s − 3·11-s − 510·17-s + 459·19-s − 144·23-s + 22·25-s + 570·29-s + 2.64e3·31-s + 2.73e3·35-s + 433·37-s − 98·43-s + 1.77e3·47-s − 1.12e3·49-s + 213·53-s + 117·55-s + 4.85e3·59-s − 1.29e4·61-s + 7.20e3·67-s − 1.91e4·71-s + 1.43e4·73-s + 210·77-s + 1.34e4·79-s + 1.98e4·85-s + 3.61e4·89-s − 1.79e4·95-s + 234·101-s + ⋯ |
| L(s) = 1 | − 1.55·5-s − 1.42·7-s − 0.0247·11-s − 1.76·17-s + 1.27·19-s − 0.272·23-s + 0.0351·25-s + 0.677·29-s + 2.74·31-s + 2.22·35-s + 0.316·37-s − 0.0530·43-s + 0.801·47-s − 0.469·49-s + 0.0758·53-s + 0.0386·55-s + 1.39·59-s − 3.47·61-s + 1.60·67-s − 3.80·71-s + 2.69·73-s + 0.0354·77-s + 2.15·79-s + 2.75·85-s + 4.56·89-s − 1.98·95-s + 0.0229·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.338498643\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.338498643\) |
| \(L(3)\) |
|
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 | $C_2$ | \( ( 1 + 5 p T + p^{4} T^{2} )^{2} \) |
| good | 5 | $D_4\times C_2$ | \( 1 + 39 T + 1499 T^{2} + 38688 T^{3} + 910314 T^{4} + 38688 p^{4} T^{5} + 1499 p^{8} T^{6} + 39 p^{12} T^{7} + p^{16} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 3 T - 22991 T^{2} - 18846 T^{3} + 314509350 T^{4} - 18846 p^{4} T^{5} - 22991 p^{8} T^{6} + 3 p^{12} T^{7} + p^{16} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - 104545 T^{2} + 4347712992 T^{4} - 104545 p^{8} T^{6} + p^{16} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 + 30 p T + 273878 T^{2} + 5615340 p T^{3} + 35301116943 T^{4} + 5615340 p^{5} T^{5} + 273878 p^{8} T^{6} + 30 p^{13} T^{7} + p^{16} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 459 T + 167807 T^{2} - 44789220 T^{3} + 1690346226 T^{4} - 44789220 p^{4} T^{5} + 167807 p^{8} T^{6} - 459 p^{12} T^{7} + p^{16} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 144 T - 511298 T^{2} - 3981312 T^{3} + 198946408899 T^{4} - 3981312 p^{4} T^{5} - 511298 p^{8} T^{6} + 144 p^{12} T^{7} + p^{16} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 285 T + 957650 T^{2} - 285 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 2640 T + 4626383 T^{2} - 6080403120 T^{3} + 6597284881248 T^{4} - 6080403120 p^{4} T^{5} + 4626383 p^{8} T^{6} - 2640 p^{12} T^{7} + p^{16} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 433 T - 3370571 T^{2} + 82383446 T^{3} + 8795926937950 T^{4} + 82383446 p^{4} T^{5} - 3370571 p^{8} T^{6} - 433 p^{12} T^{7} + p^{16} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 1911628 T^{2} - 5103203602650 T^{4} - 1911628 p^{8} T^{6} + p^{16} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 49 T + 6504624 T^{2} + 49 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 1770 T + 2863406 T^{2} - 3219817620 T^{3} - 15406289154225 T^{4} - 3219817620 p^{4} T^{5} + 2863406 p^{8} T^{6} - 1770 p^{12} T^{7} + p^{16} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 213 T + 10571119 T^{2} + 5603329656 T^{3} + 48173739418650 T^{4} + 5603329656 p^{4} T^{5} + 10571119 p^{8} T^{6} - 213 p^{12} T^{7} + p^{16} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 4857 T + 11344973 T^{2} - 16909596930 T^{3} - 39425002755858 T^{4} - 16909596930 p^{4} T^{5} + 11344973 p^{8} T^{6} - 4857 p^{12} T^{7} + p^{16} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 12936 T + 97241618 T^{2} + 536347076496 T^{3} + 2299677254685027 T^{4} + 536347076496 p^{4} T^{5} + 97241618 p^{8} T^{6} + 12936 p^{12} T^{7} + p^{16} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 7205 T + 16284235 T^{2} + 33679426660 T^{3} - 182813754402566 T^{4} + 33679426660 p^{4} T^{5} + 16284235 p^{8} T^{6} - 7205 p^{12} T^{7} + p^{16} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 9594 T + 73513946 T^{2} + 9594 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 14355 T + 111552041 T^{2} - 615303618930 T^{3} + 2981445599576550 T^{4} - 615303618930 p^{4} T^{5} + 111552041 p^{8} T^{6} - 14355 p^{12} T^{7} + p^{16} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 13424 T + 77161687 T^{2} - 337505228048 T^{3} + 2081611623871504 T^{4} - 337505228048 p^{4} T^{5} + 77161687 p^{8} T^{6} - 13424 p^{12} T^{7} + p^{16} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 139759861 T^{2} + 8846001164179884 T^{4} - 139759861 p^{8} T^{6} + p^{16} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 36162 T + 666660278 T^{2} - 8344870771860 T^{3} + 76664356786491087 T^{4} - 8344870771860 p^{4} T^{5} + 666660278 p^{8} T^{6} - 36162 p^{12} T^{7} + p^{16} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 320265193 T^{2} + 41187945645458400 T^{4} - 320265193 p^{8} T^{6} + p^{16} T^{8} \) |
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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
−7.977735360325764868998053843511, −7.973700757272200269425419759544, −7.70709288761037991194328471283, −7.22693385849840865763061291038, −6.98766707881181502240027620466, −6.80376355668477288610478598129, −6.51438539842388610562779778427, −6.24294419416518263523235920621, −6.07905351013507296871098161313, −5.81364838167430681429799610996, −5.20515159992305447037699378501, −4.97478270493992642467904329605, −4.63576258160137496369315497203, −4.48638354443723451364677445654, −3.96040683658174259712922973289, −3.90179887329594313091401945997, −3.53766629981009816423388965347, −3.03960084691929810823464138426, −2.81023466967138783149866794817, −2.71676928907771292469222258830, −1.96792552166365922925382980923, −1.68178434070739740953667460755, −0.75498782061535669825088591362, −0.70345892617689995640790888784, −0.27442906615369269688982964783,
0.27442906615369269688982964783, 0.70345892617689995640790888784, 0.75498782061535669825088591362, 1.68178434070739740953667460755, 1.96792552166365922925382980923, 2.71676928907771292469222258830, 2.81023466967138783149866794817, 3.03960084691929810823464138426, 3.53766629981009816423388965347, 3.90179887329594313091401945997, 3.96040683658174259712922973289, 4.48638354443723451364677445654, 4.63576258160137496369315497203, 4.97478270493992642467904329605, 5.20515159992305447037699378501, 5.81364838167430681429799610996, 6.07905351013507296871098161313, 6.24294419416518263523235920621, 6.51438539842388610562779778427, 6.80376355668477288610478598129, 6.98766707881181502240027620466, 7.22693385849840865763061291038, 7.70709288761037991194328471283, 7.973700757272200269425419759544, 7.977735360325764868998053843511
