| L(s) = 1 | − 4·4-s + 8·5-s + 10·16-s − 32·20-s + 36·25-s + 16·37-s − 16·41-s − 16·43-s + 16·47-s + 48·59-s − 20·64-s − 32·67-s + 80·80-s − 16·83-s + 80·89-s − 144·100-s + 64·101-s + 48·109-s + 32·121-s + 120·125-s + 127-s + 131-s + 137-s + 139-s − 64·148-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 2·4-s + 3.57·5-s + 5/2·16-s − 7.15·20-s + 36/5·25-s + 2.63·37-s − 2.49·41-s − 2.43·43-s + 2.33·47-s + 6.24·59-s − 5/2·64-s − 3.90·67-s + 8.94·80-s − 1.75·83-s + 8.47·89-s − 14.3·100-s + 6.36·101-s + 4.59·109-s + 2.90·121-s + 10.7·125-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 5^{8} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 5^{8} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(14.82855635\) |
| \(L(\frac12)\) |
\(\approx\) |
\(14.82855635\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + T^{2} )^{4} \) |
| 3 | \( 1 \) |
| 5 | \( ( 1 - T )^{8} \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 - 32 T^{2} + 544 T^{4} - 8352 T^{6} + 109154 T^{8} - 8352 p^{2} T^{10} + 544 p^{4} T^{12} - 32 p^{6} T^{14} + p^{8} T^{16} \) |
| 13 | \( 1 - 80 T^{2} + 2996 T^{4} - 69104 T^{6} + 1078470 T^{8} - 69104 p^{2} T^{10} + 2996 p^{4} T^{12} - 80 p^{6} T^{14} + p^{8} T^{16} \) |
| 17 | \( ( 1 + 60 T^{2} - 8 T^{3} + 1460 T^{4} - 8 p T^{5} + 60 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( 1 - 48 T^{2} + 1092 T^{4} - 20496 T^{6} + 392870 T^{8} - 20496 p^{2} T^{10} + 1092 p^{4} T^{12} - 48 p^{6} T^{14} + p^{8} T^{16} \) |
| 23 | \( 1 - 144 T^{2} + 9428 T^{4} - 376368 T^{6} + 10282534 T^{8} - 376368 p^{2} T^{10} + 9428 p^{4} T^{12} - 144 p^{6} T^{14} + p^{8} T^{16} \) |
| 29 | \( 1 - 112 T^{2} + 7488 T^{4} - 339056 T^{6} + 11412770 T^{8} - 339056 p^{2} T^{10} + 7488 p^{4} T^{12} - 112 p^{6} T^{14} + p^{8} T^{16} \) |
| 31 | \( 1 - 176 T^{2} + 14816 T^{4} - 791216 T^{6} + 29218434 T^{8} - 791216 p^{2} T^{10} + 14816 p^{4} T^{12} - 176 p^{6} T^{14} + p^{8} T^{16} \) |
| 37 | \( ( 1 - 8 T + 52 T^{2} - 432 T^{3} + 3764 T^{4} - 432 p T^{5} + 52 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 + 8 T + 116 T^{2} + 664 T^{3} + 6526 T^{4} + 664 p T^{5} + 116 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 + 8 T + 124 T^{2} + 720 T^{3} + 7508 T^{4} + 720 p T^{5} + 124 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 - 8 T + 156 T^{2} - 880 T^{3} + 10404 T^{4} - 880 p T^{5} + 156 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 53 | \( 1 - 248 T^{2} + 34028 T^{4} - 3010504 T^{6} + 189188806 T^{8} - 3010504 p^{2} T^{10} + 34028 p^{4} T^{12} - 248 p^{6} T^{14} + p^{8} T^{16} \) |
| 59 | \( ( 1 - 24 T + 432 T^{2} - 4904 T^{3} + 44786 T^{4} - 4904 p T^{5} + 432 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 61 | \( 1 - 432 T^{2} + 84740 T^{4} - 9833616 T^{6} + 737501350 T^{8} - 9833616 p^{2} T^{10} + 84740 p^{4} T^{12} - 432 p^{6} T^{14} + p^{8} T^{16} \) |
| 67 | \( ( 1 + 16 T + 276 T^{2} + 2936 T^{3} + 27492 T^{4} + 2936 p T^{5} + 276 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 71 | \( 1 - 416 T^{2} + 80900 T^{4} - 9781984 T^{6} + 820517062 T^{8} - 9781984 p^{2} T^{10} + 80900 p^{4} T^{12} - 416 p^{6} T^{14} + p^{8} T^{16} \) |
| 73 | \( 1 - 248 T^{2} + 27212 T^{4} - 23368 p T^{6} + 97521382 T^{8} - 23368 p^{3} T^{10} + 27212 p^{4} T^{12} - 248 p^{6} T^{14} + p^{8} T^{16} \) |
| 79 | \( ( 1 + 188 T^{2} + 19270 T^{4} + 188 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( ( 1 + 8 T + 68 T^{2} + 360 T^{3} + 7766 T^{4} + 360 p T^{5} + 68 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 - 40 T + 876 T^{2} - 12824 T^{3} + 139590 T^{4} - 12824 p T^{5} + 876 p^{2} T^{6} - 40 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 - 316 T^{2} + 43270 T^{4} - 316 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.48761825089128845727966119396, −3.23409348615497216831998865974, −3.13343470253420420776354516719, −3.13094565643052889993189194736, −3.07473190847235927188455557440, −2.81511955001201688710829999272, −2.77399699848389249446743844236, −2.65145298822757771154554395375, −2.50103823453450004411243309506, −2.18771997443452214850038639488, −2.09715010833598922238676932033, −2.05170491729270560503428551283, −2.00633637091141325652434008054, −1.88776649621781265367959235602, −1.87363805689378556160022972578, −1.80425569095442498014734538162, −1.58050931293641078697651764689, −1.43693853852274283329739730043, −0.896272727382262917578629968620, −0.841565372131464261073062773118, −0.829619874333495730260006858687, −0.78882543013543125447112669155, −0.74100620111524828707310295158, −0.63971385935018173210675191552, −0.14970251639041850358147173280,
0.14970251639041850358147173280, 0.63971385935018173210675191552, 0.74100620111524828707310295158, 0.78882543013543125447112669155, 0.829619874333495730260006858687, 0.841565372131464261073062773118, 0.896272727382262917578629968620, 1.43693853852274283329739730043, 1.58050931293641078697651764689, 1.80425569095442498014734538162, 1.87363805689378556160022972578, 1.88776649621781265367959235602, 2.00633637091141325652434008054, 2.05170491729270560503428551283, 2.09715010833598922238676932033, 2.18771997443452214850038639488, 2.50103823453450004411243309506, 2.65145298822757771154554395375, 2.77399699848389249446743844236, 2.81511955001201688710829999272, 3.07473190847235927188455557440, 3.13094565643052889993189194736, 3.13343470253420420776354516719, 3.23409348615497216831998865974, 3.48761825089128845727966119396
Plot not available for L-functions of degree greater than 10.
