| L(s) = 1 | + 8·4-s + 16·7-s − 8·13-s + 42·16-s + 40·19-s + 44·25-s + 128·28-s − 56·31-s + 136·37-s − 104·43-s − 116·49-s − 64·52-s − 8·61-s + 136·64-s + 112·67-s + 448·73-s + 320·76-s + 448·79-s − 128·91-s − 152·97-s + 352·100-s − 104·103-s − 680·109-s + 672·112-s − 44·121-s − 448·124-s + 127-s + ⋯ |
| L(s) = 1 | + 2·4-s + 16/7·7-s − 0.615·13-s + 21/8·16-s + 2.10·19-s + 1.75·25-s + 32/7·28-s − 1.80·31-s + 3.67·37-s − 2.41·43-s − 2.36·49-s − 1.23·52-s − 0.131·61-s + 17/8·64-s + 1.67·67-s + 6.13·73-s + 4.21·76-s + 5.67·79-s − 1.40·91-s − 1.56·97-s + 3.51·100-s − 1.00·103-s − 6.23·109-s + 6·112-s − 0.363·121-s − 3.61·124-s + 0.00787·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(10.86090261\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.86090261\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( ( 1 + p T^{2} )^{4} \) |
| good | 2 | \( 1 - p^{3} T^{2} + 11 p T^{4} + 3 p^{3} T^{6} - 375 T^{8} + 3 p^{7} T^{10} + 11 p^{9} T^{12} - p^{15} T^{14} + p^{16} T^{16} \) |
| 5 | \( 1 - 44 T^{2} + 2896 T^{4} - 15972 p T^{6} + 2805486 T^{8} - 15972 p^{5} T^{10} + 2896 p^{8} T^{12} - 44 p^{12} T^{14} + p^{16} T^{16} \) |
| 7 | \( ( 1 - 8 T + 22 p T^{2} - 160 p T^{3} + 10526 T^{4} - 160 p^{3} T^{5} + 22 p^{5} T^{6} - 8 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 13 | \( ( 1 + 4 T + 22 p T^{2} - 388 T^{3} + 42374 T^{4} - 388 p^{2} T^{5} + 22 p^{5} T^{6} + 4 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 17 | \( 1 - 1232 T^{2} + 735100 T^{4} - 289134384 T^{6} + 90323356422 T^{8} - 289134384 p^{4} T^{10} + 735100 p^{8} T^{12} - 1232 p^{12} T^{14} + p^{16} T^{16} \) |
| 19 | \( ( 1 - 20 T + 868 T^{2} - 8916 T^{3} + 328146 T^{4} - 8916 p^{2} T^{5} + 868 p^{4} T^{6} - 20 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 23 | \( 1 - 2444 T^{2} + 3254416 T^{4} - 2852397780 T^{6} + 1774428987822 T^{8} - 2852397780 p^{4} T^{10} + 3254416 p^{8} T^{12} - 2444 p^{12} T^{14} + p^{16} T^{16} \) |
| 29 | \( 1 - 1880 T^{2} + 1924252 T^{4} - 2074330856 T^{6} + 2128273704070 T^{8} - 2074330856 p^{4} T^{10} + 1924252 p^{8} T^{12} - 1880 p^{12} T^{14} + p^{16} T^{16} \) |
| 31 | \( ( 1 + 28 T + 1432 T^{2} + 13412 T^{3} + 1149278 T^{4} + 13412 p^{2} T^{5} + 1432 p^{4} T^{6} + 28 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 37 | \( ( 1 - 68 T + 4504 T^{2} - 191964 T^{3} + 7669374 T^{4} - 191964 p^{2} T^{5} + 4504 p^{4} T^{6} - 68 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 41 | \( 1 - 8624 T^{2} + 37064956 T^{4} - 103569069648 T^{6} + 204770887955334 T^{8} - 103569069648 p^{4} T^{10} + 37064956 p^{8} T^{12} - 8624 p^{12} T^{14} + p^{16} T^{16} \) |
| 43 | \( ( 1 + 52 T + 3460 T^{2} + 22452 T^{3} + 2716146 T^{4} + 22452 p^{2} T^{5} + 3460 p^{4} T^{6} + 52 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 47 | \( 1 - 6620 T^{2} + 27373072 T^{4} - 80988531396 T^{6} + 198687369383598 T^{8} - 80988531396 p^{4} T^{10} + 27373072 p^{8} T^{12} - 6620 p^{12} T^{14} + p^{16} T^{16} \) |
| 53 | \( 1 - 14348 T^{2} + 100839952 T^{4} - 466253289236 T^{6} + 1539595638444910 T^{8} - 466253289236 p^{4} T^{10} + 100839952 p^{8} T^{12} - 14348 p^{12} T^{14} + p^{16} T^{16} \) |
| 59 | \( 1 - 19976 T^{2} + 192056668 T^{4} - 1159676388024 T^{6} + 4810763889113478 T^{8} - 1159676388024 p^{4} T^{10} + 192056668 p^{8} T^{12} - 19976 p^{12} T^{14} + p^{16} T^{16} \) |
| 61 | \( ( 1 + 4 T + 8686 T^{2} + 150828 T^{3} + 38662374 T^{4} + 150828 p^{2} T^{5} + 8686 p^{4} T^{6} + 4 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 67 | \( ( 1 - 56 T + 15436 T^{2} - 560264 T^{3} + 95435590 T^{4} - 560264 p^{2} T^{5} + 15436 p^{4} T^{6} - 56 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 71 | \( 1 - 22412 T^{2} + 258098896 T^{4} - 2017610554196 T^{6} + 11687868067441198 T^{8} - 2017610554196 p^{4} T^{10} + 258098896 p^{8} T^{12} - 22412 p^{12} T^{14} + p^{16} T^{16} \) |
| 73 | \( ( 1 - 224 T + 28780 T^{2} - 2677024 T^{3} + 212472614 T^{4} - 2677024 p^{2} T^{5} + 28780 p^{4} T^{6} - 224 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 79 | \( ( 1 - 224 T + 40282 T^{2} - 4437320 T^{3} + 5320162 p T^{4} - 4437320 p^{2} T^{5} + 40282 p^{4} T^{6} - 224 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 83 | \( 1 - 26528 T^{2} + 386566684 T^{4} - 3936191194208 T^{6} + 30559983176028934 T^{8} - 3936191194208 p^{4} T^{10} + 386566684 p^{8} T^{12} - 26528 p^{12} T^{14} + p^{16} T^{16} \) |
| 89 | \( 1 - 13664 T^{2} + 265530628 T^{4} - 2460680918048 T^{6} + 25256672766991750 T^{8} - 2460680918048 p^{4} T^{10} + 265530628 p^{8} T^{12} - 13664 p^{12} T^{14} + p^{16} T^{16} \) |
| 97 | \( ( 1 + 76 T + 22576 T^{2} + 814804 T^{3} + 238699630 T^{4} + 814804 p^{2} T^{5} + 22576 p^{4} T^{6} + 76 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
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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
−6.43582175903541550346058559994, −6.16492224891397776236856513674, −6.12239081148757597872577383111, −5.51666761476179908773503919558, −5.40649851662024829122327852253, −5.37375796084896131956724437492, −5.16861813201072160803308590498, −5.11511974884193308201814717549, −5.00896290831815675914722534385, −4.76445427698419078254017929745, −4.46461980900513337316974775279, −4.40982244585631747860046535376, −3.79492696364512226621247730019, −3.76791304914603876587605377418, −3.64566846317129982146168578188, −3.30269668537560490287510187510, −3.01584346191154152023953725264, −2.93963427071291783872155317884, −2.53155448642799038955089705164, −2.37018323971967355982322231806, −1.95146832998413090276851987671, −1.85412047241642429588746034994, −1.50825432679160940571810485086, −1.09294384407139854265682891105, −0.823160333623156396346803876693,
0.823160333623156396346803876693, 1.09294384407139854265682891105, 1.50825432679160940571810485086, 1.85412047241642429588746034994, 1.95146832998413090276851987671, 2.37018323971967355982322231806, 2.53155448642799038955089705164, 2.93963427071291783872155317884, 3.01584346191154152023953725264, 3.30269668537560490287510187510, 3.64566846317129982146168578188, 3.76791304914603876587605377418, 3.79492696364512226621247730019, 4.40982244585631747860046535376, 4.46461980900513337316974775279, 4.76445427698419078254017929745, 5.00896290831815675914722534385, 5.11511974884193308201814717549, 5.16861813201072160803308590498, 5.37375796084896131956724437492, 5.40649851662024829122327852253, 5.51666761476179908773503919558, 6.12239081148757597872577383111, 6.16492224891397776236856513674, 6.43582175903541550346058559994
Plot not available for L-functions of degree greater than 10.
