| L(s) = 1 | + 4·2-s + 8·4-s − 8·5-s + 4·7-s + 8·8-s + 6·9-s − 32·10-s + 20·11-s + 16·14-s − 4·16-s + 24·18-s − 20·19-s − 64·20-s + 80·22-s + 32·25-s + 32·28-s + 48·29-s − 20·31-s − 32·32-s − 32·35-s + 48·36-s + 48·37-s − 80·38-s − 64·40-s − 120·41-s + 160·44-s − 48·45-s + ⋯ |
| L(s) = 1 | + 2·2-s + 2·4-s − 8/5·5-s + 4/7·7-s + 8-s + 2/3·9-s − 3.19·10-s + 1.81·11-s + 8/7·14-s − 1/4·16-s + 4/3·18-s − 1.05·19-s − 3.19·20-s + 3.63·22-s + 1.27·25-s + 8/7·28-s + 1.65·29-s − 0.645·31-s − 32-s − 0.914·35-s + 4/3·36-s + 1.29·37-s − 2.10·38-s − 8/5·40-s − 2.92·41-s + 3.63·44-s − 1.06·45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(5.999101715\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.999101715\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( ( 1 - p T + p T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | | \( 1 \) |
| good | 5 | $D_4\times C_2$ | \( 1 + 8 T + 32 T^{2} + 72 T^{3} - 94 T^{4} + 72 p^{2} T^{5} + 32 p^{4} T^{6} + 8 p^{6} T^{7} + p^{8} T^{8} \) |
| 7 | $C_2^2$ | \( ( 1 - 2 T + 2 T^{2} - 2 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 - 20 T + 200 T^{2} - 2940 T^{3} + 42542 T^{4} - 2940 p^{2} T^{5} + 200 p^{4} T^{6} - 20 p^{6} T^{7} + p^{8} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 260 T^{2} + 107142 T^{4} - 260 p^{4} T^{6} + p^{8} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 20 T + 200 T^{2} + 6300 T^{3} + 196334 T^{4} + 6300 p^{2} T^{5} + 200 p^{4} T^{6} + 20 p^{6} T^{7} + p^{8} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 452 T^{2} - 80442 T^{4} - 452 p^{4} T^{6} + p^{8} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 24 T + 626 T^{2} - 24 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 + 20 T + 200 T^{2} + 2940 T^{3} - 619378 T^{4} + 2940 p^{2} T^{5} + 200 p^{4} T^{6} + 20 p^{6} T^{7} + p^{8} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 48 T + 1152 T^{2} + 35664 T^{3} - 3356446 T^{4} + 35664 p^{2} T^{5} + 1152 p^{4} T^{6} - 48 p^{6} T^{7} + p^{8} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 + 120 T + 7200 T^{2} + 345720 T^{3} + 15160322 T^{4} + 345720 p^{2} T^{5} + 7200 p^{4} T^{6} + 120 p^{6} T^{7} + p^{8} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 5060 T^{2} + 12009702 T^{4} - 5060 p^{4} T^{6} + p^{8} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 12 T + 72 T^{2} - 19524 T^{3} + 4955534 T^{4} - 19524 p^{2} T^{5} + 72 p^{4} T^{6} - 12 p^{6} T^{7} + p^{8} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 + 20 T + 918 T^{2} + 20 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 + 100 T + 5000 T^{2} - 66900 T^{3} - 16327378 T^{4} - 66900 p^{2} T^{5} + 5000 p^{4} T^{6} + 100 p^{6} T^{7} + p^{8} T^{8} \) |
| 61 | $C_2$ | \( ( 1 + 46 T + p^{2} T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 126 T + 7938 T^{2} - 126 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 + 60 T + 1800 T^{2} - 432300 T^{3} - 46140466 T^{4} - 432300 p^{2} T^{5} + 1800 p^{4} T^{6} + 60 p^{6} T^{7} + p^{8} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 + 16 T + 128 T^{2} - 67824 T^{3} - 55608382 T^{4} - 67824 p^{2} T^{5} + 128 p^{4} T^{6} + 16 p^{6} T^{7} + p^{8} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 56 T + 12066 T^{2} + 56 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 + 60 T + 1800 T^{2} + 116340 T^{3} - 16983058 T^{4} + 116340 p^{2} T^{5} + 1800 p^{4} T^{6} + 60 p^{6} T^{7} + p^{8} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 + 40 T + 800 T^{2} + 47400 T^{3} - 42565246 T^{4} + 47400 p^{2} T^{5} + 800 p^{4} T^{6} + 40 p^{6} T^{7} + p^{8} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 288 T + 41472 T^{2} - 5004576 T^{3} + 540431234 T^{4} - 5004576 p^{2} T^{5} + 41472 p^{4} T^{6} - 288 p^{6} T^{7} + p^{8} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.92518599996390021862865629856, −6.78980263577284125789712060664, −6.25301534048139235224179163400, −6.16363032479543272200557752586, −6.06160548857750352002590962203, −5.90121143417884513587874582472, −5.61329002323184789516427958234, −4.97843773869478712801234986268, −4.87735606354748328339680432935, −4.67805069914261202141218658186, −4.59530399647482849913383058145, −4.51039608691683200086657592870, −4.18501323627660341961825161030, −3.89214782985566787543263985496, −3.52127879797911566263852512030, −3.50525234667647535635977399706, −3.28225882820726605503852227614, −3.02692170121247205378828774327, −2.66362540477710162622553994322, −2.00028453942906745218469622054, −1.93189775836751995558640163087, −1.72701770280239962473009582508, −1.04036413759809795519752033507, −0.873535400542340948380626758931, −0.24215637976086381025199741096,
0.24215637976086381025199741096, 0.873535400542340948380626758931, 1.04036413759809795519752033507, 1.72701770280239962473009582508, 1.93189775836751995558640163087, 2.00028453942906745218469622054, 2.66362540477710162622553994322, 3.02692170121247205378828774327, 3.28225882820726605503852227614, 3.50525234667647535635977399706, 3.52127879797911566263852512030, 3.89214782985566787543263985496, 4.18501323627660341961825161030, 4.51039608691683200086657592870, 4.59530399647482849913383058145, 4.67805069914261202141218658186, 4.87735606354748328339680432935, 4.97843773869478712801234986268, 5.61329002323184789516427958234, 5.90121143417884513587874582472, 6.06160548857750352002590962203, 6.16363032479543272200557752586, 6.25301534048139235224179163400, 6.78980263577284125789712060664, 6.92518599996390021862865629856
