| L(s) = 1 | − 8·3-s + 4-s + 36·9-s − 8·12-s − 6·13-s + 2·16-s + 20·17-s − 6·23-s + 3·25-s − 120·27-s − 18·29-s + 36·36-s + 48·39-s + 34·43-s − 16·48-s − 4·49-s − 160·51-s − 6·52-s − 10·53-s − 20·61-s + 10·64-s + 20·68-s + 48·69-s − 24·75-s − 2·79-s + 330·81-s + 144·87-s + ⋯ |
| L(s) = 1 | − 4.61·3-s + 1/2·4-s + 12·9-s − 2.30·12-s − 1.66·13-s + 1/2·16-s + 4.85·17-s − 1.25·23-s + 3/5·25-s − 23.0·27-s − 3.34·29-s + 6·36-s + 7.68·39-s + 5.18·43-s − 2.30·48-s − 4/7·49-s − 22.4·51-s − 0.832·52-s − 1.37·53-s − 2.56·61-s + 5/4·64-s + 2.42·68-s + 5.77·69-s − 2.77·75-s − 0.225·79-s + 36.6·81-s + 15.4·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5968578922\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5968578922\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( ( 1 + T )^{8} \) |
| 7 | \( ( 1 + T^{2} )^{4} \) |
| 13 | \( 1 + 6 T - 6 p T^{3} - 306 T^{4} - 6 p^{2} T^{5} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| good | 2 | \( 1 - T^{2} - T^{4} - 7 T^{6} + 3 p^{3} T^{8} - 7 p^{2} T^{10} - p^{4} T^{12} - p^{6} T^{14} + p^{8} T^{16} \) |
| 5 | \( 1 - 3 T^{2} + 58 T^{4} - 49 p T^{6} + 1754 T^{8} - 49 p^{3} T^{10} + 58 p^{4} T^{12} - 3 p^{6} T^{14} + p^{8} T^{16} \) |
| 11 | \( 1 - 4 p T^{2} + 1060 T^{4} - 17700 T^{6} + 221750 T^{8} - 17700 p^{2} T^{10} + 1060 p^{4} T^{12} - 4 p^{7} T^{14} + p^{8} T^{16} \) |
| 17 | \( ( 1 - 10 T + 76 T^{2} - 398 T^{3} + 1846 T^{4} - 398 p T^{5} + 76 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 19 | \( 1 - 55 T^{2} + 1034 T^{4} + 1119 T^{6} - 294198 T^{8} + 1119 p^{2} T^{10} + 1034 p^{4} T^{12} - 55 p^{6} T^{14} + p^{8} T^{16} \) |
| 23 | \( ( 1 + 3 T + 14 T^{2} + 67 T^{3} + 938 T^{4} + 67 p T^{5} + 14 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 + 9 T + 102 T^{2} + 539 T^{3} + 3794 T^{4} + 539 p T^{5} + 102 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 31 | \( 1 - 159 T^{2} + 12386 T^{4} - 630417 T^{6} + 22910554 T^{8} - 630417 p^{2} T^{10} + 12386 p^{4} T^{12} - 159 p^{6} T^{14} + p^{8} T^{16} \) |
| 37 | \( 1 - 120 T^{2} + 8476 T^{4} - 437896 T^{6} + 17867558 T^{8} - 437896 p^{2} T^{10} + 8476 p^{4} T^{12} - 120 p^{6} T^{14} + p^{8} T^{16} \) |
| 41 | \( 1 - 80 T^{2} + 3436 T^{4} - 131616 T^{6} + 6401414 T^{8} - 131616 p^{2} T^{10} + 3436 p^{4} T^{12} - 80 p^{6} T^{14} + p^{8} T^{16} \) |
| 43 | \( ( 1 - 17 T + 236 T^{2} - 2033 T^{3} + 15702 T^{4} - 2033 p T^{5} + 236 p^{2} T^{6} - 17 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 47 | \( 1 - 135 T^{2} + 12874 T^{4} - 792137 T^{6} + 42595610 T^{8} - 792137 p^{2} T^{10} + 12874 p^{4} T^{12} - 135 p^{6} T^{14} + p^{8} T^{16} \) |
| 53 | \( ( 1 + 5 T + 70 T^{2} + 439 T^{3} + 2514 T^{4} + 439 p T^{5} + 70 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 59 | \( 1 - 224 T^{2} + 28060 T^{4} - 2524560 T^{6} + 171683654 T^{8} - 2524560 p^{2} T^{10} + 28060 p^{4} T^{12} - 224 p^{6} T^{14} + p^{8} T^{16} \) |
| 61 | \( ( 1 + 10 T + 216 T^{2} + 1694 T^{3} + 19230 T^{4} + 1694 p T^{5} + 216 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 67 | \( 1 - 84 T^{2} + 11060 T^{4} - 557964 T^{6} + 51760822 T^{8} - 557964 p^{2} T^{10} + 11060 p^{4} T^{12} - 84 p^{6} T^{14} + p^{8} T^{16} \) |
| 71 | \( 1 - 524 T^{2} + 122980 T^{4} - 16877220 T^{6} + 1480949270 T^{8} - 16877220 p^{2} T^{10} + 122980 p^{4} T^{12} - 524 p^{6} T^{14} + p^{8} T^{16} \) |
| 73 | \( 1 - 279 T^{2} + 38006 T^{4} - 3345129 T^{6} + 248830834 T^{8} - 3345129 p^{2} T^{10} + 38006 p^{4} T^{12} - 279 p^{6} T^{14} + p^{8} T^{16} \) |
| 79 | \( ( 1 + T + 284 T^{2} + 133 T^{3} + 32310 T^{4} + 133 p T^{5} + 284 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 83 | \( 1 - 423 T^{2} + 91858 T^{4} - 12813617 T^{6} + 1256995274 T^{8} - 12813617 p^{2} T^{10} + 91858 p^{4} T^{12} - 423 p^{6} T^{14} + p^{8} T^{16} \) |
| 89 | \( 1 - 459 T^{2} + 105746 T^{4} - 15824037 T^{6} + 1663278058 T^{8} - 15824037 p^{2} T^{10} + 105746 p^{4} T^{12} - 459 p^{6} T^{14} + p^{8} T^{16} \) |
| 97 | \( 1 - 223 T^{2} + 28838 T^{4} - 2060433 T^{6} + 169926930 T^{8} - 2060433 p^{2} T^{10} + 28838 p^{4} T^{12} - 223 p^{6} T^{14} + p^{8} T^{16} \) |
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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
−5.58414330373274965584556389889, −5.46574611784435309914614765565, −5.15731459349275679588299606724, −4.89788523456000695994481810766, −4.86374818263898927334881126369, −4.69741478460604261659142356811, −4.69181573237370921403565320532, −4.43744846945057481022249463891, −4.11542342142085977188427012875, −4.09932386311301587863076699306, −3.90085870543605897686305727419, −3.84971803648037961104394204622, −3.44824586229474762633637821250, −3.21293486685804373081758743109, −3.14672796628683823510044043828, −3.00273709224930792780520742132, −2.89364023055856544602029483310, −2.12247920713668402735662001322, −1.92936211941682309606075020867, −1.92744139719366819874944200990, −1.89991757754696464948889841600, −1.13444297621586274594223265677, −1.13382989259055541112227494903, −0.61110081856408412835972233806, −0.58587456943888545848658042567,
0.58587456943888545848658042567, 0.61110081856408412835972233806, 1.13382989259055541112227494903, 1.13444297621586274594223265677, 1.89991757754696464948889841600, 1.92744139719366819874944200990, 1.92936211941682309606075020867, 2.12247920713668402735662001322, 2.89364023055856544602029483310, 3.00273709224930792780520742132, 3.14672796628683823510044043828, 3.21293486685804373081758743109, 3.44824586229474762633637821250, 3.84971803648037961104394204622, 3.90085870543605897686305727419, 4.09932386311301587863076699306, 4.11542342142085977188427012875, 4.43744846945057481022249463891, 4.69181573237370921403565320532, 4.69741478460604261659142356811, 4.86374818263898927334881126369, 4.89788523456000695994481810766, 5.15731459349275679588299606724, 5.46574611784435309914614765565, 5.58414330373274965584556389889
Plot not available for L-functions of degree greater than 10.
