| L(s) = 1 | − 4·3-s + 16·7-s + 18·9-s + 8·13-s + 8·19-s − 64·21-s − 44·27-s − 120·31-s − 8·37-s − 32·39-s − 328·43-s − 36·49-s − 32·57-s + 8·61-s + 288·63-s + 152·67-s − 32·73-s − 88·79-s + 170·81-s + 128·91-s + 480·93-s − 144·97-s − 144·103-s − 216·109-s + 32·111-s + 144·117-s + 80·121-s + ⋯ |
| L(s) = 1 | − 4/3·3-s + 16/7·7-s + 2·9-s + 8/13·13-s + 8/19·19-s − 3.04·21-s − 1.62·27-s − 3.87·31-s − 0.216·37-s − 0.820·39-s − 7.62·43-s − 0.734·49-s − 0.561·57-s + 8/61·61-s + 32/7·63-s + 2.26·67-s − 0.438·73-s − 1.11·79-s + 2.09·81-s + 1.40·91-s + 5.16·93-s − 1.48·97-s − 1.39·103-s − 1.98·109-s + 0.288·111-s + 1.23·117-s + 0.661·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.04473356684\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.04473356684\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 4 T - 2 T^{2} - 4 p^{2} T^{3} - 34 p T^{4} - 4 p^{4} T^{5} - 2 p^{4} T^{6} + 4 p^{6} T^{7} + p^{8} T^{8} \) |
| 5 | \( 1 \) |
| good | 7 | \( ( 1 - 8 T + 114 T^{2} - 304 T^{3} + 4426 T^{4} - 304 p^{2} T^{5} + 114 p^{4} T^{6} - 8 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 11 | \( 1 - 80 T^{2} - 8612 T^{4} + 1315920 T^{6} + 21445638 T^{8} + 1315920 p^{4} T^{10} - 8612 p^{8} T^{12} - 80 p^{12} T^{14} + p^{16} T^{16} \) |
| 13 | \( ( 1 - 4 T + 296 T^{2} + 1620 T^{3} + 39470 T^{4} + 1620 p^{2} T^{5} + 296 p^{4} T^{6} - 4 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 17 | \( 1 - 1208 T^{2} + 736924 T^{4} - 304675080 T^{6} + 97385661510 T^{8} - 304675080 p^{4} T^{10} + 736924 p^{8} T^{12} - 1208 p^{12} T^{14} + p^{16} T^{16} \) |
| 19 | \( ( 1 - 4 T + 736 T^{2} - 908 T^{3} + 287486 T^{4} - 908 p^{2} T^{5} + 736 p^{4} T^{6} - 4 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 23 | \( 1 - 1268 T^{2} + 1124824 T^{4} - 670011900 T^{6} + 391776576750 T^{8} - 670011900 p^{4} T^{10} + 1124824 p^{8} T^{12} - 1268 p^{12} T^{14} + p^{16} T^{16} \) |
| 29 | \( 1 - 4352 T^{2} + 8975164 T^{4} - 11861548800 T^{6} + 11435522898630 T^{8} - 11861548800 p^{4} T^{10} + 8975164 p^{8} T^{12} - 4352 p^{12} T^{14} + p^{16} T^{16} \) |
| 31 | \( ( 1 + 60 T + 2920 T^{2} + 101620 T^{3} + 3613902 T^{4} + 101620 p^{2} T^{5} + 2920 p^{4} T^{6} + 60 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 37 | \( ( 1 + 4 T + 2248 T^{2} - 23316 T^{3} + 2375598 T^{4} - 23316 p^{2} T^{5} + 2248 p^{4} T^{6} + 4 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 41 | \( 1 - 1920 T^{2} + 9436924 T^{4} - 13365676160 T^{6} + 37834324951686 T^{8} - 13365676160 p^{4} T^{10} + 9436924 p^{8} T^{12} - 1920 p^{12} T^{14} + p^{16} T^{16} \) |
| 43 | \( ( 1 + 164 T + 16878 T^{2} + 1130284 T^{3} + 57159898 T^{4} + 1130284 p^{2} T^{5} + 16878 p^{4} T^{6} + 164 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 47 | \( 1 - 15060 T^{2} + 103552984 T^{4} - 425803156380 T^{6} + 1147648197554286 T^{8} - 425803156380 p^{4} T^{10} + 103552984 p^{8} T^{12} - 15060 p^{12} T^{14} + p^{16} T^{16} \) |
| 53 | \( 1 - 5496 T^{2} + 16907740 T^{4} - 66116431304 T^{6} + 230534565871494 T^{8} - 66116431304 p^{4} T^{10} + 16907740 p^{8} T^{12} - 5496 p^{12} T^{14} + p^{16} T^{16} \) |
| 59 | \( 1 - 15232 T^{2} + 125199548 T^{4} - 692185732224 T^{6} + 2791710943429190 T^{8} - 692185732224 p^{4} T^{10} + 125199548 p^{8} T^{12} - 15232 p^{12} T^{14} + p^{16} T^{16} \) |
| 61 | \( ( 1 - 4 T + 2376 T^{2} - 32108 T^{3} + 20621806 T^{4} - 32108 p^{2} T^{5} + 2376 p^{4} T^{6} - 4 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 67 | \( ( 1 - 76 T + 13102 T^{2} - 787268 T^{3} + 76659290 T^{4} - 787268 p^{2} T^{5} + 13102 p^{4} T^{6} - 76 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 71 | \( 1 - 24024 T^{2} + 302202556 T^{4} - 2503345638248 T^{6} + 14778735683172486 T^{8} - 2503345638248 p^{4} T^{10} + 302202556 p^{8} T^{12} - 24024 p^{12} T^{14} + p^{16} T^{16} \) |
| 73 | \( ( 1 + 16 T + 17788 T^{2} + 226416 T^{3} + 135726918 T^{4} + 226416 p^{2} T^{5} + 17788 p^{4} T^{6} + 16 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 79 | \( ( 1 + 44 T + 23696 T^{2} + 832068 T^{3} + 217860926 T^{4} + 832068 p^{2} T^{5} + 23696 p^{4} T^{6} + 44 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 83 | \( 1 - 13396 T^{2} + 137438936 T^{4} - 1239330480732 T^{6} + 8621185666547246 T^{8} - 1239330480732 p^{4} T^{10} + 137438936 p^{8} T^{12} - 13396 p^{12} T^{14} + p^{16} T^{16} \) |
| 89 | \( 1 - 14056 T^{2} + 62829020 T^{4} + 214903235496 T^{6} - 3856000255517626 T^{8} + 214903235496 p^{4} T^{10} + 62829020 p^{8} T^{12} - 14056 p^{12} T^{14} + p^{16} T^{16} \) |
| 97 | \( ( 1 + 72 T + 27260 T^{2} + 1496952 T^{3} + 346190214 T^{4} + 1496952 p^{2} T^{5} + 27260 p^{4} T^{6} + 72 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
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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
−3.97143373521287650025092893608, −3.69229933525715734161955668734, −3.62525903305469021588687420467, −3.53174872664939624530941930878, −3.42938618515720240728178950562, −3.42506157615034625801458846735, −3.39906540223322574100606659914, −3.12353016205193154549720237508, −2.86311113744821102096216861219, −2.74582469023170673780311208237, −2.36913915301760801065942612228, −2.30713014251499384853853600446, −2.18214405105012144347760406153, −2.09124442245925906429052609781, −2.02156572510541823525350677449, −1.57997211056449361047463984777, −1.51578314056836495300853496257, −1.48270256334560020027324843798, −1.37894235330521263508730252439, −1.21221949460811362331580289792, −1.10921646335036839843212271484, −1.05215836953936899795533512131, −0.28595368234134848833708724313, −0.12268138814292972874944522677, −0.06696985084232075874188191739,
0.06696985084232075874188191739, 0.12268138814292972874944522677, 0.28595368234134848833708724313, 1.05215836953936899795533512131, 1.10921646335036839843212271484, 1.21221949460811362331580289792, 1.37894235330521263508730252439, 1.48270256334560020027324843798, 1.51578314056836495300853496257, 1.57997211056449361047463984777, 2.02156572510541823525350677449, 2.09124442245925906429052609781, 2.18214405105012144347760406153, 2.30713014251499384853853600446, 2.36913915301760801065942612228, 2.74582469023170673780311208237, 2.86311113744821102096216861219, 3.12353016205193154549720237508, 3.39906540223322574100606659914, 3.42506157615034625801458846735, 3.42938618515720240728178950562, 3.53174872664939624530941930878, 3.62525903305469021588687420467, 3.69229933525715734161955668734, 3.97143373521287650025092893608
Plot not available for L-functions of degree greater than 10.
