| L(s) = 1 | + 2·5-s − 7-s − 11-s + 13-s + 4·17-s + 4·19-s + 5·23-s + 25-s − 29-s − 5·31-s − 2·35-s + 24·37-s + 12·41-s − 2·43-s − 13·47-s + 6·49-s − 6·53-s − 2·55-s + 10·59-s − 14·61-s + 2·65-s − 10·67-s + 6·71-s + 36·73-s + 77-s − 6·79-s − 20·83-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 0.377·7-s − 0.301·11-s + 0.277·13-s + 0.970·17-s + 0.917·19-s + 1.04·23-s + 1/5·25-s − 0.185·29-s − 0.898·31-s − 0.338·35-s + 3.94·37-s + 1.87·41-s − 0.304·43-s − 1.89·47-s + 6/7·49-s − 0.824·53-s − 0.269·55-s + 1.30·59-s − 1.79·61-s + 0.248·65-s − 1.22·67-s + 0.712·71-s + 4.21·73-s + 0.113·77-s − 0.675·79-s − 2.19·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{16} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{16} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.436342120\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.436342120\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| good | 7 | $D_4\times C_2$ | \( 1 + T - 5 T^{2} - 8 T^{3} - 20 T^{4} - 8 p T^{5} - 5 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + T - 13 T^{2} - 8 T^{3} + 64 T^{4} - 8 p T^{5} - 13 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - T - 17 T^{2} + 8 T^{3} + 142 T^{4} + 8 p T^{5} - 17 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )^{4} \) |
| 23 | $D_4\times C_2$ | \( 1 - 5 T - 19 T^{2} + 10 T^{3} + 832 T^{4} + 10 p T^{5} - 19 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 + T - 49 T^{2} - 8 T^{3} + 1630 T^{4} - 8 p T^{5} - 49 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2$$\times$$C_2^2$ | \( ( 1 + 5 T + p T^{2} )^{2}( 1 - 5 T - 6 T^{2} - 5 p T^{3} + p^{2} T^{4} ) \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 41 | $D_4\times C_2$ | \( 1 - 12 T + 59 T^{2} - 36 T^{3} - 360 T^{4} - 36 p T^{5} + 59 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 2 T - 50 T^{2} - 64 T^{3} + 895 T^{4} - 64 p T^{5} - 50 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 13 T + 41 T^{2} + 442 T^{3} + 6232 T^{4} + 442 p T^{5} + 41 p^{2} T^{6} + 13 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 + 3 T + 34 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 5 T - 34 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 + 14 T + 58 T^{2} + 224 T^{3} + 3367 T^{4} + 224 p T^{5} + 58 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 10 T - 26 T^{2} - 80 T^{3} + 4687 T^{4} - 80 p T^{5} - 26 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 3 T + 136 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 - 18 T + 194 T^{2} - 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 + 6 T - 98 T^{2} - 144 T^{3} + 8871 T^{4} - 144 p T^{5} - 98 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2^2$ | \( ( 1 + 10 T + 17 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 - 3 T + 172 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 + 2 T - 158 T^{2} - 64 T^{3} + 16447 T^{4} - 64 p T^{5} - 158 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−6.02252127372296267759680190717, −5.98208856926853479526461508057, −5.83353693111733644051869985713, −5.54105862305303005650400098581, −5.35588359540097145314319695253, −5.23706705582916443126165243521, −4.76410206688183916965380006555, −4.69608712032358468060580860442, −4.67692977067778736597453657844, −4.40710131673675750102138561264, −3.93021461278635452995412584924, −3.87908493669760562453600206740, −3.60148448259644478562198771181, −3.23919982772485728603187505649, −3.13493548238853908723297514829, −3.11879955324003492350427039399, −2.57703439589641857720868267979, −2.39379911977339480225420684982, −2.33345830853821436962046102734, −2.00340359272702501072677485001, −1.55121773493558672883958471402, −1.18736198595815857656106018395, −1.17321888833871824271831428920, −0.66615072709403905147191067825, −0.46400530550641295319183729903,
0.46400530550641295319183729903, 0.66615072709403905147191067825, 1.17321888833871824271831428920, 1.18736198595815857656106018395, 1.55121773493558672883958471402, 2.00340359272702501072677485001, 2.33345830853821436962046102734, 2.39379911977339480225420684982, 2.57703439589641857720868267979, 3.11879955324003492350427039399, 3.13493548238853908723297514829, 3.23919982772485728603187505649, 3.60148448259644478562198771181, 3.87908493669760562453600206740, 3.93021461278635452995412584924, 4.40710131673675750102138561264, 4.67692977067778736597453657844, 4.69608712032358468060580860442, 4.76410206688183916965380006555, 5.23706705582916443126165243521, 5.35588359540097145314319695253, 5.54105862305303005650400098581, 5.83353693111733644051869985713, 5.98208856926853479526461508057, 6.02252127372296267759680190717
