L(s) = 1 | − 4·3-s + 12·5-s + 8·9-s − 4·11-s + 4·13-s − 48·15-s − 20·17-s + 40·23-s + 72·25-s − 16·27-s − 48·31-s + 16·33-s + 104·37-s − 16·39-s + 16·41-s − 188·43-s + 96·45-s + 84·47-s + 80·51-s + 4·53-s − 48·55-s − 288·61-s + 48·65-s + 92·67-s − 160·69-s + 160·71-s − 72·73-s + ⋯ |
L(s) = 1 | − 4/3·3-s + 12/5·5-s + 8/9·9-s − 0.363·11-s + 4/13·13-s − 3.19·15-s − 1.17·17-s + 1.73·23-s + 2.87·25-s − 0.592·27-s − 1.54·31-s + 0.484·33-s + 2.81·37-s − 0.410·39-s + 0.390·41-s − 4.37·43-s + 2.13·45-s + 1.78·47-s + 1.56·51-s + 4/53·53-s − 0.872·55-s − 4.72·61-s + 0.738·65-s + 1.37·67-s − 2.31·69-s + 2.25·71-s − 0.986·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1492750191\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1492750191\) |
\(L(2)\) |
|
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 | | \( 1 \) |
| 5 | $C_2^2$ | \( 1 - 12 T + 72 T^{2} - 12 p^{2} T^{3} + p^{4} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
good | 3 | $D_4\times C_2$ | \( 1 + 4 T + 8 T^{2} + 16 T^{3} + 7 T^{4} + 16 p^{2} T^{5} + 8 p^{4} T^{6} + 4 p^{6} T^{7} + p^{8} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 + 2 T + 19 T^{2} + 2 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 4 T + 8 T^{2} - 432 T^{3} + 19607 T^{4} - 432 p^{2} T^{5} + 8 p^{4} T^{6} - 4 p^{6} T^{7} + p^{8} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 + 20 T + 200 T^{2} - 80 T^{3} - 85817 T^{4} - 80 p^{2} T^{5} + 200 p^{4} T^{6} + 20 p^{6} T^{7} + p^{8} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 768 T^{2} + 389954 T^{4} - 768 p^{4} T^{6} + p^{8} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 40 T + 800 T^{2} - 1160 T^{3} - 248318 T^{4} - 1160 p^{2} T^{5} + 800 p^{4} T^{6} - 40 p^{6} T^{7} + p^{8} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 466 T^{2} + 371251 T^{4} - 466 p^{4} T^{6} + p^{8} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 + 24 T + 1940 T^{2} + 24 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 104 T + 5408 T^{2} - 256776 T^{3} + 10981922 T^{4} - 256776 p^{2} T^{5} + 5408 p^{4} T^{6} - 104 p^{6} T^{7} + p^{8} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 8 T + 2244 T^{2} - 8 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 + 188 T + 17672 T^{2} + 1157140 T^{3} + 57226414 T^{4} + 1157140 p^{2} T^{5} + 17672 p^{4} T^{6} + 188 p^{6} T^{7} + p^{8} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 84 T + 3528 T^{2} + 51408 T^{3} - 7208953 T^{4} + 51408 p^{2} T^{5} + 3528 p^{4} T^{6} - 84 p^{6} T^{7} + p^{8} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 4 T + 8 T^{2} - 8444 T^{3} + 8425438 T^{4} - 8444 p^{2} T^{5} + 8 p^{4} T^{6} - 4 p^{6} T^{7} + p^{8} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 7592 T^{2} + 38442738 T^{4} - 7592 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 144 T + 12402 T^{2} + 144 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 92 T + 4232 T^{2} - 445924 T^{3} + 46858654 T^{4} - 445924 p^{2} T^{5} + 4232 p^{4} T^{6} - 92 p^{6} T^{7} + p^{8} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 80 T + 11626 T^{2} - 80 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 72 T + 2592 T^{2} + 35208 T^{3} - 22947358 T^{4} + 35208 p^{2} T^{5} + 2592 p^{4} T^{6} + 72 p^{6} T^{7} + p^{8} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 8274 T^{2} + 26761235 T^{4} - 8274 p^{4} T^{6} + p^{8} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 + 256 T + 32768 T^{2} + 3746048 T^{3} + 368279842 T^{4} + 3746048 p^{2} T^{5} + 32768 p^{4} T^{6} + 256 p^{6} T^{7} + p^{8} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 - 9442 T^{2} + p^{4} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 + 84 T + 3528 T^{2} + 652176 T^{3} + 117853367 T^{4} + 652176 p^{2} T^{5} + 3528 p^{4} T^{6} + 84 p^{6} T^{7} + p^{8} T^{8} \) |
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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
−7.39115346942530162240969638512, −7.34136161504796369249286205430, −6.73441426906295551406433470773, −6.71303396276242405520828867273, −6.63217034189318587148121570772, −6.25648609009628062767509683851, −6.10727404617646850868192114141, −5.86843991960686648062093013544, −5.46483076016824383455292584165, −5.45197282552488988399320268003, −5.14210742580949871221718358404, −5.13694003362421802305930987104, −4.43814710747988390368157338718, −4.41559626442155562024721796072, −4.31956337958299848648031086830, −3.65622553733123518398735044868, −3.27556964346086226171161004792, −2.90527081189509733556571835856, −2.78305653022852903720279351364, −2.31344564300988376083276667639, −1.89469312700939811994581843911, −1.74543241384198270338334617889, −1.18538970030321454223862449574, −1.04616846480562040634261300688, −0.07147302759002677953156355029,
0.07147302759002677953156355029, 1.04616846480562040634261300688, 1.18538970030321454223862449574, 1.74543241384198270338334617889, 1.89469312700939811994581843911, 2.31344564300988376083276667639, 2.78305653022852903720279351364, 2.90527081189509733556571835856, 3.27556964346086226171161004792, 3.65622553733123518398735044868, 4.31956337958299848648031086830, 4.41559626442155562024721796072, 4.43814710747988390368157338718, 5.13694003362421802305930987104, 5.14210742580949871221718358404, 5.45197282552488988399320268003, 5.46483076016824383455292584165, 5.86843991960686648062093013544, 6.10727404617646850868192114141, 6.25648609009628062767509683851, 6.63217034189318587148121570772, 6.71303396276242405520828867273, 6.73441426906295551406433470773, 7.34136161504796369249286205430, 7.39115346942530162240969638512