| L(s) = 1 | + 8·9-s + 104·25-s + 112·29-s − 208·37-s + 4·49-s + 240·53-s + 84·81-s − 304·113-s − 472·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 936·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | + 8/9·9-s + 4.15·25-s + 3.86·29-s − 5.62·37-s + 4/49·49-s + 4.52·53-s + 1.03·81-s − 2.69·113-s − 3.90·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 5.53·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + 0.00473·211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(20.09336180\) |
| \(L(\frac12)\) |
\(\approx\) |
\(20.09336180\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - 4 T^{2} + 38 p^{2} T^{4} - 4 p^{4} T^{6} + p^{8} T^{8} \) |
| good | 3 | \( ( 1 - 4 T^{2} - 2 p^{2} T^{4} - 4 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 5 | \( ( 1 - 52 T^{2} + 1742 T^{4} - 52 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 11 | \( ( 1 + 236 T^{2} + 31430 T^{4} + 236 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 13 | \( ( 1 - 36 p T^{2} + 102862 T^{4} - 36 p^{5} T^{6} + p^{8} T^{8} )^{2} \) |
| 17 | \( ( 1 - 4 T^{2} + 155270 T^{4} - 4 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 19 | \( ( 1 - 1188 T^{2} + 613294 T^{4} - 1188 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 23 | \( ( 1 + 1676 T^{2} + 1214822 T^{4} + 1676 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 29 | \( ( 1 - 28 T + 1142 T^{2} - 28 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 31 | \( ( 1 - 2180 T^{2} + 2458118 T^{4} - 2180 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 37 | \( ( 1 + 52 T + 2678 T^{2} + 52 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 41 | \( ( 1 - 4868 T^{2} + 10725062 T^{4} - 4868 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 43 | \( ( 1 + 2540 T^{2} + 7272902 T^{4} + 2540 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 47 | \( ( 1 - 6788 T^{2} + 21266822 T^{4} - 6788 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 53 | \( ( 1 - 30 T + p^{2} T^{2} )^{8} \) |
| 59 | \( ( 1 - 4580 T^{2} + 29037038 T^{4} - 4580 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 61 | \( ( 1 - 4212 T^{2} + 3996814 T^{4} - 4212 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 67 | \( ( 1 + 8150 T^{2} + p^{4} T^{4} )^{4} \) |
| 71 | \( ( 1 + 11964 T^{2} + 70047686 T^{4} + 11964 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 73 | \( ( 1 - 8388 T^{2} + 67882822 T^{4} - 8388 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 79 | \( ( 1 - 1028 T^{2} + 37863942 T^{4} - 1028 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 83 | \( ( 1 - 8132 T^{2} + 106317422 T^{4} - 8132 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 89 | \( ( 1 - 30916 T^{2} + 364289990 T^{4} - 30916 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 97 | \( ( 1 - 19204 T^{2} + 268303110 T^{4} - 19204 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
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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
−4.71664336950341982886961754340, −4.61266215396720601723518585367, −4.47457097411464470465854651849, −4.21460578209618555530799735897, −4.12218479482438654444026324752, −3.76730826423391282411168820318, −3.74933764164083268151851285461, −3.63607460104509761848056084906, −3.58073443980855305138723091982, −3.14545883813259510549431807322, −3.09498573371229128613998896116, −3.09063356594634539198035096771, −2.75180132977316826928167585303, −2.61121509677837890644900077459, −2.56279088805458355110054179458, −2.42512572653450738557814564749, −1.91906023409882922214957639818, −1.86630804721481802246516301696, −1.67140672544115934542138631581, −1.41309078315179452410358457763, −1.36325695417813051180722616317, −0.816921724682102397314624067953, −0.71000803673743955149061749778, −0.57831500456113735993924844069, −0.50169817016792925087487846483,
0.50169817016792925087487846483, 0.57831500456113735993924844069, 0.71000803673743955149061749778, 0.816921724682102397314624067953, 1.36325695417813051180722616317, 1.41309078315179452410358457763, 1.67140672544115934542138631581, 1.86630804721481802246516301696, 1.91906023409882922214957639818, 2.42512572653450738557814564749, 2.56279088805458355110054179458, 2.61121509677837890644900077459, 2.75180132977316826928167585303, 3.09063356594634539198035096771, 3.09498573371229128613998896116, 3.14545883813259510549431807322, 3.58073443980855305138723091982, 3.63607460104509761848056084906, 3.74933764164083268151851285461, 3.76730826423391282411168820318, 4.12218479482438654444026324752, 4.21460578209618555530799735897, 4.47457097411464470465854651849, 4.61266215396720601723518585367, 4.71664336950341982886961754340
Plot not available for L-functions of degree greater than 10.
