| L(s) = 1 | − 4·3-s + 8·9-s + 96·13-s − 16·19-s + 12·27-s + 72·31-s − 112·37-s − 384·39-s + 240·43-s + 360·49-s + 64·57-s − 208·61-s + 232·67-s − 136·79-s − 34·81-s − 288·93-s − 480·97-s − 64·109-s + 448·111-s + 768·117-s + 127-s − 960·129-s + 131-s + 137-s + 139-s − 1.44e3·147-s + 149-s + ⋯ |
| L(s) = 1 | − 4/3·3-s + 8/9·9-s + 7.38·13-s − 0.842·19-s + 4/9·27-s + 2.32·31-s − 3.02·37-s − 9.84·39-s + 5.58·43-s + 7.34·49-s + 1.12·57-s − 3.40·61-s + 3.46·67-s − 1.72·79-s − 0.419·81-s − 3.09·93-s − 4.94·97-s − 0.587·109-s + 4.03·111-s + 6.56·117-s + 0.00787·127-s − 7.44·129-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s − 9.79·147-s + 0.00671·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{56} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{56} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(15.78196251\) |
| \(L(\frac12)\) |
\(\approx\) |
\(15.78196251\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 4 T + 8 T^{2} - 4 p T^{3} - 14 p^{2} T^{4} - 4 p^{3} T^{5} + 8 p^{4} T^{6} + 4 p^{6} T^{7} + p^{8} T^{8} \) |
| good | 5 | \( 1 + 372 T^{4} + 464614 T^{8} + 372 p^{8} T^{12} + p^{16} T^{16} \) |
| 7 | \( ( 1 - 180 T^{2} + 12874 T^{4} - 180 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 11 | \( 1 + 37380 T^{4} + 692870854 T^{8} + 37380 p^{8} T^{12} + p^{16} T^{16} \) |
| 13 | \( ( 1 - 48 T + 1152 T^{2} - 21264 T^{3} + 317422 T^{4} - 21264 p^{2} T^{5} + 1152 p^{4} T^{6} - 48 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 17 | \( ( 1 - 236 T^{2} + 73334 T^{4} - 236 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 19 | \( ( 1 + 8 T + 32 T^{2} + 1160 T^{3} - 4606 T^{4} + 1160 p^{2} T^{5} + 32 p^{4} T^{6} + 8 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 23 | \( ( 1 + 996 T^{2} + 719878 T^{4} + 996 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 29 | \( 1 + 147060 T^{4} + 623812106854 T^{8} + 147060 p^{8} T^{12} + p^{16} T^{16} \) |
| 31 | \( ( 1 - 18 T + 1996 T^{2} - 18 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 37 | \( ( 1 + 56 T + 1568 T^{2} + 79016 T^{3} + 3980078 T^{4} + 79016 p^{2} T^{5} + 1568 p^{4} T^{6} + 56 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 41 | \( ( 1 + 6060 T^{2} + 14724790 T^{4} + 6060 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 43 | \( ( 1 - 120 T + 7200 T^{2} - 411000 T^{3} + 20977474 T^{4} - 411000 p^{2} T^{5} + 7200 p^{4} T^{6} - 120 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 47 | \( ( 1 - 692 T^{2} + 7491686 T^{4} - 692 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 53 | \( 1 + 1441524 T^{4} - 91064986479386 T^{8} + 1441524 p^{8} T^{12} + p^{16} T^{16} \) |
| 59 | \( 1 + 15787044 T^{4} + 346992256453126 T^{8} + 15787044 p^{8} T^{12} + p^{16} T^{16} \) |
| 61 | \( ( 1 + 104 T + 5408 T^{2} + 491192 T^{3} + 43609454 T^{4} + 491192 p^{2} T^{5} + 5408 p^{4} T^{6} + 104 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 67 | \( ( 1 - 116 T + 6728 T^{2} - 350436 T^{3} + 16097858 T^{4} - 350436 p^{2} T^{5} + 6728 p^{4} T^{6} - 116 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 71 | \( ( 1 + 3604 T^{2} + 10180454 T^{4} + 3604 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 73 | \( ( 1 - 17604 T^{2} + 131615494 T^{4} - 17604 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 79 | \( ( 1 + 34 T + 11588 T^{2} + 34 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 83 | \( 1 - 64624380 T^{4} + 2364945173919814 T^{8} - 64624380 p^{8} T^{12} + p^{16} T^{16} \) |
| 89 | \( ( 1 + 25444 T^{2} + 277784198 T^{4} + 25444 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 97 | \( ( 1 + 120 T + 21970 T^{2} + 120 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
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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
−5.00579883504016950251990549912, −4.33155677022440545507988724840, −4.27117322926042117478366909885, −4.18857128215640756086530404426, −4.14262176404159059512732154272, −4.09293244439004174608171956476, −4.07017583473527598462663020204, −3.73271571081024151529926470469, −3.62438303496762471810317013131, −3.62220533689448611069439870348, −3.51354836201917330484158650388, −2.99203110125751283456903362659, −2.74477896380149761912823471855, −2.69974637061842803290563608221, −2.59114828121415343592799496696, −2.57425309045472827535512621903, −2.18491369627934028763623282697, −1.67782246281251738608200624864, −1.44203153880854921599146985678, −1.42668546379484756089983495971, −1.30416726334722289148493871782, −1.05332531094639369396394099107, −0.78452393529266808689949882403, −0.64477962522212284094104441036, −0.47632914028539930661677401497,
0.47632914028539930661677401497, 0.64477962522212284094104441036, 0.78452393529266808689949882403, 1.05332531094639369396394099107, 1.30416726334722289148493871782, 1.42668546379484756089983495971, 1.44203153880854921599146985678, 1.67782246281251738608200624864, 2.18491369627934028763623282697, 2.57425309045472827535512621903, 2.59114828121415343592799496696, 2.69974637061842803290563608221, 2.74477896380149761912823471855, 2.99203110125751283456903362659, 3.51354836201917330484158650388, 3.62220533689448611069439870348, 3.62438303496762471810317013131, 3.73271571081024151529926470469, 4.07017583473527598462663020204, 4.09293244439004174608171956476, 4.14262176404159059512732154272, 4.18857128215640756086530404426, 4.27117322926042117478366909885, 4.33155677022440545507988724840, 5.00579883504016950251990549912
Plot not available for L-functions of degree greater than 10.
