| L(s) = 1 | + 4·4-s + 32·13-s + 4·16-s + 160·19-s + 16·25-s − 16·31-s + 304·37-s − 80·43-s − 14·49-s + 128·52-s − 232·61-s − 16·64-s + 192·67-s − 192·73-s + 640·76-s − 304·79-s + 288·97-s + 64·100-s − 272·103-s − 576·109-s − 20·121-s − 64·124-s + 127-s + 131-s + 137-s + 139-s + 1.21e3·148-s + ⋯ |
| L(s) = 1 | + 4-s + 2.46·13-s + 1/4·16-s + 8.42·19-s + 0.639·25-s − 0.516·31-s + 8.21·37-s − 1.86·43-s − 2/7·49-s + 2.46·52-s − 3.80·61-s − 1/4·64-s + 2.86·67-s − 2.63·73-s + 8.42·76-s − 3.84·79-s + 2.96·97-s + 0.639·100-s − 2.64·103-s − 5.28·109-s − 0.165·121-s − 0.516·124-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 8.21·148-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{32} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{32} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(85.27446469\) |
| \(L(\frac12)\) |
\(\approx\) |
\(85.27446469\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 3 | \( 1 \) |
| 7 | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| good | 5 | \( 1 - 16 T^{2} - 122 p T^{4} + 6144 T^{6} + 199331 T^{8} + 6144 p^{4} T^{10} - 122 p^{9} T^{12} - 16 p^{12} T^{14} + p^{16} T^{16} \) |
| 11 | \( 1 + 20 T^{2} - 21814 T^{4} - 141360 T^{6} + 273246515 T^{8} - 141360 p^{4} T^{10} - 21814 p^{8} T^{12} + 20 p^{12} T^{14} + p^{16} T^{16} \) |
| 13 | \( ( 1 - 16 T - 34 T^{2} + 768 T^{3} + 12275 T^{4} + 768 p^{2} T^{5} - 34 p^{4} T^{6} - 16 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 17 | \( ( 1 - 368 T^{2} + 125186 T^{4} - 368 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 19 | \( ( 1 - 20 T + p^{2} T^{2} )^{8} \) |
| 23 | \( 1 + 1652 T^{2} + 1494314 T^{4} + 1115278416 T^{6} + 682680166355 T^{8} + 1115278416 p^{4} T^{10} + 1494314 p^{8} T^{12} + 1652 p^{12} T^{14} + p^{16} T^{16} \) |
| 29 | \( 1 + 1472 T^{2} + 857438 T^{4} - 154877952 T^{6} - 318384619549 T^{8} - 154877952 p^{4} T^{10} + 857438 p^{8} T^{12} + 1472 p^{12} T^{14} + p^{16} T^{16} \) |
| 31 | \( ( 1 + 8 T - 46 p T^{2} - 3456 T^{3} + 1235075 T^{4} - 3456 p^{2} T^{5} - 46 p^{5} T^{6} + 8 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 37 | \( ( 1 - 38 T + p^{2} T^{2} )^{8} \) |
| 41 | \( 1 + 2800 T^{2} + 3167806 T^{4} - 2742118400 T^{6} - 8309494695485 T^{8} - 2742118400 p^{4} T^{10} + 3167806 p^{8} T^{12} + 2800 p^{12} T^{14} + p^{16} T^{16} \) |
| 43 | \( ( 1 + 40 T + 1534 T^{2} - 145280 T^{3} - 6372845 T^{4} - 145280 p^{2} T^{5} + 1534 p^{4} T^{6} + 40 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 47 | \( ( 1 + 4130 T^{2} + 12177219 T^{4} + 4130 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 53 | \( ( 1 + 4928 T^{2} + 21271650 T^{4} + 4928 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 59 | \( 1 + 10532 T^{2} + 61824746 T^{4} + 261862971792 T^{6} + 919716139941299 T^{8} + 261862971792 p^{4} T^{10} + 61824746 p^{8} T^{12} + 10532 p^{12} T^{14} + p^{16} T^{16} \) |
| 61 | \( ( 1 + 116 T + 4442 T^{2} + 182352 T^{3} + 17336579 T^{4} + 182352 p^{2} T^{5} + 4442 p^{4} T^{6} + 116 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 67 | \( ( 1 - 96 T + 5102 T^{2} + 466944 T^{3} - 44596749 T^{4} + 466944 p^{2} T^{5} + 5102 p^{4} T^{6} - 96 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 71 | \( ( 1 - 19700 T^{2} + 147838694 T^{4} - 19700 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 73 | \( ( 1 + 48 T + 8434 T^{2} + 48 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 79 | \( ( 1 + 152 T + 6638 T^{2} + 605568 T^{3} + 87987011 T^{4} + 605568 p^{2} T^{5} + 6638 p^{4} T^{6} + 152 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 83 | \( 1 + 9380 T^{2} + 431786 T^{4} - 69074582640 T^{6} + 1237576087901555 T^{8} - 69074582640 p^{4} T^{10} + 431786 p^{8} T^{12} + 9380 p^{12} T^{14} + p^{16} T^{16} \) |
| 89 | \( ( 1 - 12208 T^{2} + 68358210 T^{4} - 12208 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 97 | \( ( 1 - 144 T + 10286 T^{2} + 1204992 T^{3} - 174431805 T^{4} + 1204992 p^{2} T^{5} + 10286 p^{4} T^{6} - 144 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
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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
−4.02563555060268598043930092539, −3.75097800675615115826273140882, −3.63128406596415461472847613455, −3.52661284838254713035230324895, −3.24622199403334279921853970746, −3.10775372763943176427501568292, −3.07940277746527040369680129912, −3.05741416085387142968091723842, −3.03810757380775073073465340116, −3.00934169814642374883538701326, −2.76233669969794867741667310890, −2.46011659367349382835240112355, −2.39104611257922351719781810833, −2.35640020101182696521952406877, −1.84175596836477599046318131089, −1.69157879732901682335398933332, −1.53049619262773107684491039646, −1.51818351425576642480727469984, −1.31019503050528938947396341197, −1.10038652688314184322251737233, −0.959551412215547924656435109649, −0.924993363380909363129768379114, −0.795405736925728765070821887066, −0.50318410622064081490544760211, −0.37306348615541258959608951117,
0.37306348615541258959608951117, 0.50318410622064081490544760211, 0.795405736925728765070821887066, 0.924993363380909363129768379114, 0.959551412215547924656435109649, 1.10038652688314184322251737233, 1.31019503050528938947396341197, 1.51818351425576642480727469984, 1.53049619262773107684491039646, 1.69157879732901682335398933332, 1.84175596836477599046318131089, 2.35640020101182696521952406877, 2.39104611257922351719781810833, 2.46011659367349382835240112355, 2.76233669969794867741667310890, 3.00934169814642374883538701326, 3.03810757380775073073465340116, 3.05741416085387142968091723842, 3.07940277746527040369680129912, 3.10775372763943176427501568292, 3.24622199403334279921853970746, 3.52661284838254713035230324895, 3.63128406596415461472847613455, 3.75097800675615115826273140882, 4.02563555060268598043930092539
Plot not available for L-functions of degree greater than 10.
