| L(s) = 1 | − 4·3-s − 6·7-s + 11·9-s + 8·11-s − 12·13-s + 8·17-s + 24·21-s + 4·23-s − 20·27-s − 18·29-s + 16·31-s − 32·33-s + 6·37-s + 48·39-s − 4·41-s + 28·43-s + 11·49-s − 32·51-s + 20·53-s + 24·59-s + 2·61-s − 66·63-s + 18·67-s − 16·69-s + 8·71-s + 24·73-s − 48·77-s + ⋯ |
| L(s) = 1 | − 2.30·3-s − 2.26·7-s + 11/3·9-s + 2.41·11-s − 3.32·13-s + 1.94·17-s + 5.23·21-s + 0.834·23-s − 3.84·27-s − 3.34·29-s + 2.87·31-s − 5.57·33-s + 0.986·37-s + 7.68·39-s − 0.624·41-s + 4.26·43-s + 11/7·49-s − 4.48·51-s + 2.74·53-s + 3.12·59-s + 0.256·61-s − 8.31·63-s + 2.19·67-s − 1.92·69-s + 0.949·71-s + 2.80·73-s − 5.47·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.494589075\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.494589075\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 13 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| good | 3 | $D_4\times C_2$ | \( 1 + 4 T + 5 T^{2} - 4 T^{3} - 20 T^{4} - 4 p T^{5} + 5 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 + 6 T + 25 T^{2} + 78 T^{3} + 204 T^{4} + 78 p T^{5} + 25 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 8 T + 17 T^{2} + 8 p T^{3} - 52 p T^{4} + 8 p^{2} T^{5} + 17 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 8 T + 41 T^{2} - 164 T^{3} + 628 T^{4} - 164 p T^{5} + 41 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 9 T^{2} + 96 T^{3} - 124 T^{4} + 96 p T^{5} + 9 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 4 T + 53 T^{2} - 244 T^{3} + 1588 T^{4} - 244 p T^{5} + 53 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 + 18 T + 177 T^{2} + 1242 T^{3} + 7052 T^{4} + 1242 p T^{5} + 177 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 16 T + 128 T^{2} - 912 T^{3} + 5822 T^{4} - 912 p T^{5} + 128 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 6 T + 85 T^{2} - 438 T^{3} + 4404 T^{4} - 438 p T^{5} + 85 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 + 4 T + 53 T^{2} + 424 T^{3} + 2092 T^{4} + 424 p T^{5} + 53 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 28 T + 365 T^{2} - 3060 T^{3} + 20876 T^{4} - 3060 p T^{5} + 365 p^{2} T^{6} - 28 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 132 T^{2} + 8006 T^{4} - 132 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 20 T + 200 T^{2} - 1580 T^{3} + 11806 T^{4} - 1580 p T^{5} + 200 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 24 T + 153 T^{2} + 1176 T^{3} - 21292 T^{4} + 1176 p T^{5} + 153 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2}( 1 + 13 T + p T^{2} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 18 T + 121 T^{2} - 1242 T^{3} + 15012 T^{4} - 1242 p T^{5} + 121 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 8 T + 41 T^{2} - 272 T^{3} - 1748 T^{4} - 272 p T^{5} + 41 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 12 T + 134 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 - 164 T^{2} + 16134 T^{4} - 164 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $C_2^2$ | \( ( 1 - 154 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 28 T + 197 T^{2} + 2408 T^{3} - 48788 T^{4} + 2408 p T^{5} + 197 p^{2} T^{6} - 28 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 2 T - 179 T^{2} + 22 T^{3} + 23692 T^{4} + 22 p T^{5} - 179 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−6.92531356498586834594752233137, −6.56454312347139173860545133418, −6.47588285234550885081584999640, −6.42711715778246116329766043206, −6.01382517793420824736925898841, −5.61518955941947737368383292993, −5.59898337339426222436416122728, −5.48692138703710455907962805136, −5.35415682004052495437078582619, −4.83900088794655565949423611644, −4.61758241471623841922046260701, −4.59925732960216870554615254772, −4.22976954565868685737244794722, −3.75968386436507509998024796370, −3.67089760901643102967082369699, −3.65950547999373006432359031285, −3.40231329252299899968277041360, −2.62870579177595165628263388708, −2.49226641549696068668726998388, −2.33068990079707755171376170022, −2.10971480225005478917739279516, −1.17878983034632481734052860033, −1.00184314704626977858244282022, −0.67344382768063326998016782029, −0.54049740520102539802482890666,
0.54049740520102539802482890666, 0.67344382768063326998016782029, 1.00184314704626977858244282022, 1.17878983034632481734052860033, 2.10971480225005478917739279516, 2.33068990079707755171376170022, 2.49226641549696068668726998388, 2.62870579177595165628263388708, 3.40231329252299899968277041360, 3.65950547999373006432359031285, 3.67089760901643102967082369699, 3.75968386436507509998024796370, 4.22976954565868685737244794722, 4.59925732960216870554615254772, 4.61758241471623841922046260701, 4.83900088794655565949423611644, 5.35415682004052495437078582619, 5.48692138703710455907962805136, 5.59898337339426222436416122728, 5.61518955941947737368383292993, 6.01382517793420824736925898841, 6.42711715778246116329766043206, 6.47588285234550885081584999640, 6.56454312347139173860545133418, 6.92531356498586834594752233137
