| L(s) = 1 | − 8·4-s + 8·7-s − 36·13-s + 40·16-s + 72·19-s + 68·25-s − 64·28-s − 88·31-s + 108·37-s − 220·43-s − 136·49-s + 288·52-s − 160·61-s − 160·64-s − 276·67-s − 488·73-s − 576·76-s − 368·79-s − 288·91-s − 832·97-s − 544·100-s − 104·103-s − 572·109-s + 320·112-s + 704·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 2·4-s + 8/7·7-s − 2.76·13-s + 5/2·16-s + 3.78·19-s + 2.71·25-s − 2.28·28-s − 2.83·31-s + 2.91·37-s − 5.11·43-s − 2.77·49-s + 5.53·52-s − 2.62·61-s − 5/2·64-s − 4.11·67-s − 6.68·73-s − 7.57·76-s − 4.65·79-s − 3.16·91-s − 8.57·97-s − 5.43·100-s − 1.00·103-s − 5.24·109-s + 20/7·112-s + 5.67·124-s + 0.00787·127-s + 0.00763·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.001549372062\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.001549372062\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + p T^{2} )^{4} \) |
| 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 - 68 T^{2} + 1659 T^{4} + 3038 T^{6} - 780139 T^{8} + 3038 p^{4} T^{10} + 1659 p^{8} T^{12} - 68 p^{12} T^{14} + p^{16} T^{16} \) |
| 7 | \( ( 1 - 4 T + 92 T^{2} - 192 T^{3} + 5545 T^{4} - 192 p^{2} T^{5} + 92 p^{4} T^{6} - 4 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 13 | \( ( 1 + 18 T + 580 T^{2} + 8688 T^{3} + 140149 T^{4} + 8688 p^{2} T^{5} + 580 p^{4} T^{6} + 18 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 17 | \( 1 - 1114 T^{2} + 676455 T^{4} - 279227046 T^{6} + 89888383729 T^{8} - 279227046 p^{4} T^{10} + 676455 p^{8} T^{12} - 1114 p^{12} T^{14} + p^{16} T^{16} \) |
| 19 | \( ( 1 - 36 T + 1445 T^{2} - 1746 p T^{3} + 752019 T^{4} - 1746 p^{3} T^{5} + 1445 p^{4} T^{6} - 36 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 23 | \( 1 - 2890 T^{2} + 3786654 T^{4} - 3094240920 T^{6} + 1853502417991 T^{8} - 3094240920 p^{4} T^{10} + 3786654 p^{8} T^{12} - 2890 p^{12} T^{14} + p^{16} T^{16} \) |
| 29 | \( 1 - 2156 T^{2} + 3375920 T^{4} - 3473624244 T^{6} + 3944545214 p^{2} T^{8} - 3473624244 p^{4} T^{10} + 3375920 p^{8} T^{12} - 2156 p^{12} T^{14} + p^{16} T^{16} \) |
| 31 | \( ( 1 + 44 T + 3555 T^{2} + 102146 T^{3} + 4826069 T^{4} + 102146 p^{2} T^{5} + 3555 p^{4} T^{6} + 44 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 37 | \( ( 1 - 54 T + 5132 T^{2} - 196512 T^{3} + 10208045 T^{4} - 196512 p^{2} T^{5} + 5132 p^{4} T^{6} - 54 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 41 | \( 1 - 2546 T^{2} + 9721190 T^{4} - 17784166584 T^{6} + 41314150786079 T^{8} - 17784166584 p^{4} T^{10} + 9721190 p^{8} T^{12} - 2546 p^{12} T^{14} + p^{16} T^{16} \) |
| 43 | \( ( 1 + 110 T + 9636 T^{2} + 522720 T^{3} + 26258101 T^{4} + 522720 p^{2} T^{5} + 9636 p^{4} T^{6} + 110 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 47 | \( 1 - 13522 T^{2} + 85478103 T^{4} - 333215608734 T^{6} + 881858112462505 T^{8} - 333215608734 p^{4} T^{10} + 85478103 p^{8} T^{12} - 13522 p^{12} T^{14} + p^{16} T^{16} \) |
| 53 | \( 1 - 15090 T^{2} + 116674479 T^{4} - 569860816310 T^{6} + 1915316083011921 T^{8} - 569860816310 p^{4} T^{10} + 116674479 p^{8} T^{12} - 15090 p^{12} T^{14} + p^{16} T^{16} \) |
| 59 | \( 1 - 12960 T^{2} + 92791479 T^{4} - 458032464230 T^{6} + 1773271707430221 T^{8} - 458032464230 p^{4} T^{10} + 92791479 p^{8} T^{12} - 12960 p^{12} T^{14} + p^{16} T^{16} \) |
| 61 | \( ( 1 + 80 T + 14469 T^{2} + 831780 T^{3} + 80108611 T^{4} + 831780 p^{2} T^{5} + 14469 p^{4} T^{6} + 80 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 67 | \( ( 1 + 138 T + 12475 T^{2} + 776898 T^{3} + 56429749 T^{4} + 776898 p^{2} T^{5} + 12475 p^{4} T^{6} + 138 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 71 | \( 1 - 26640 T^{2} + 346358039 T^{4} - 2907307265670 T^{6} + 17242135660590621 T^{8} - 2907307265670 p^{4} T^{10} + 346358039 p^{8} T^{12} - 26640 p^{12} T^{14} + p^{16} T^{16} \) |
| 73 | \( ( 1 + 244 T + 34422 T^{2} + 3565752 T^{3} + 292613950 T^{4} + 3565752 p^{2} T^{5} + 34422 p^{4} T^{6} + 244 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 79 | \( ( 1 + 184 T + 31250 T^{2} + 3320256 T^{3} + 307843859 T^{4} + 3320256 p^{2} T^{5} + 31250 p^{4} T^{6} + 184 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 83 | \( 1 - 19194 T^{2} + 120240750 T^{4} + 162316729624 T^{6} - 5418844315837401 T^{8} + 162316729624 p^{4} T^{10} + 120240750 p^{8} T^{12} - 19194 p^{12} T^{14} + p^{16} T^{16} \) |
| 89 | \( 1 - 29790 T^{2} + 475225134 T^{4} - 5604148016720 T^{6} + 51186285493106271 T^{8} - 5604148016720 p^{4} T^{10} + 475225134 p^{8} T^{12} - 29790 p^{12} T^{14} + p^{16} T^{16} \) |
| 97 | \( ( 1 + 416 T + 97197 T^{2} + 14986998 T^{3} + 1695571135 T^{4} + 14986998 p^{2} T^{5} + 97197 p^{4} T^{6} + 416 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
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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
−3.59577844771088388271465911948, −3.44430287024083790965559707930, −3.32857748608509419439687825529, −3.31615436568613007146817837796, −3.04291884478391558964122430424, −2.98135978340665099333982956982, −2.74943458116413493925017251108, −2.73857447944533734707439476338, −2.64679284239455217568744480851, −2.59021086922616481486987239415, −2.49713716860443681886566752611, −2.45359696077673647497357818650, −1.98304557844454304610463852197, −1.57306313531569113185571454409, −1.44844619318438768004705399444, −1.40446367267347285237903390680, −1.39612927704006802087975610449, −1.38629580543496586929083664327, −1.29899028572637804239655965630, −1.25014063726346557868106862668, −1.15732236129296954273132499481, −0.36915617950077033922252468342, −0.19160752017849825094048731148, −0.05506289627161269643113012728, −0.04566420790236971380810442025,
0.04566420790236971380810442025, 0.05506289627161269643113012728, 0.19160752017849825094048731148, 0.36915617950077033922252468342, 1.15732236129296954273132499481, 1.25014063726346557868106862668, 1.29899028572637804239655965630, 1.38629580543496586929083664327, 1.39612927704006802087975610449, 1.40446367267347285237903390680, 1.44844619318438768004705399444, 1.57306313531569113185571454409, 1.98304557844454304610463852197, 2.45359696077673647497357818650, 2.49713716860443681886566752611, 2.59021086922616481486987239415, 2.64679284239455217568744480851, 2.73857447944533734707439476338, 2.74943458116413493925017251108, 2.98135978340665099333982956982, 3.04291884478391558964122430424, 3.31615436568613007146817837796, 3.32857748608509419439687825529, 3.44430287024083790965559707930, 3.59577844771088388271465911948
Plot not available for L-functions of degree greater than 10.
