| L(s) = 1 | − 8·4-s + 28·16-s + 8·25-s + 8·49-s − 72·64-s − 64·79-s − 64·100-s + 128·109-s + 160·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 80·169-s + 173-s + 179-s + 181-s + 191-s + 193-s − 64·196-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | − 4·4-s + 7·16-s + 8/5·25-s + 8/7·49-s − 9·64-s − 7.20·79-s − 6.39·100-s + 12.2·109-s + 14.5·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 6.15·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s − 4.57·196-s + 0.0712·197-s + 0.0708·199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{32} \cdot 5^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{32} \cdot 5^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1394847815\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1394847815\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( ( 1 - 4 T^{2} + 6 p T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 7 | \( ( 1 - 4 T^{2} + 6 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| good | 2 | \( ( 1 + p T^{2} + 3 T^{4} + p^{3} T^{6} + p^{4} T^{8} )^{4} \) |
| 11 | \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{8} \) |
| 13 | \( ( 1 + 20 T^{2} + 222 T^{4} + 20 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 17 | \( ( 1 - 4 T^{2} + 558 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 19 | \( ( 1 - 4 T^{2} + 510 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 23 | \( ( 1 + 56 T^{2} + 1818 T^{4} + 56 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 29 | \( ( 1 - 16 T^{2} + 1362 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 31 | \( ( 1 - 52 T^{2} + 1422 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 37 | \( ( 1 - 76 T^{2} + 3318 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 41 | \( ( 1 + 44 T^{2} + 3246 T^{4} + 44 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 43 | \( ( 1 - 100 T^{2} + 6102 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 47 | \( ( 1 - 124 T^{2} + 7398 T^{4} - 124 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 53 | \( ( 1 + 104 T^{2} + 6378 T^{4} + 104 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 59 | \( ( 1 + 92 T^{2} + 4374 T^{4} + 92 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 61 | \( ( 1 - 52 T^{2} + 6582 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 67 | \( ( 1 - 124 T^{2} + 12438 T^{4} - 124 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 71 | \( ( 1 - 184 T^{2} + 18162 T^{4} - 184 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 73 | \( ( 1 + 164 T^{2} + 13326 T^{4} + 164 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 79 | \( ( 1 + 4 T + p T^{2} )^{16} \) |
| 83 | \( ( 1 - 268 T^{2} + 30870 T^{4} - 268 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 89 | \( ( 1 + 140 T^{2} + 18798 T^{4} + 140 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 97 | \( ( 1 + 356 T^{2} + 50286 T^{4} + 356 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.39417595980536809425675077108, −3.23603288392117590536855767358, −3.22832075588624545185506941837, −3.17242810961068562627269376658, −3.03897931150445592527912206871, −2.75905827673759531140957856664, −2.72447476531906859358312338592, −2.64664613637685769588946845608, −2.59944912669301067306853146054, −2.56494558252329074986208234716, −2.37196197095986004348258665408, −2.22236277286238987950083540960, −2.00516954240082135049054479169, −1.97577303415852448482727393191, −1.96816727010849997332174532151, −1.91237660527723420637170265497, −1.58502097505968508229793693947, −1.38304850228961810857273134960, −1.27326428820631788152790280374, −1.25147645994431730424226489245, −0.985339130326871560827260802555, −0.899558821924598192617454898075, −0.67174257575544012299397704672, −0.39661742283581365434972845679, −0.11247118197453507645589789594,
0.11247118197453507645589789594, 0.39661742283581365434972845679, 0.67174257575544012299397704672, 0.899558821924598192617454898075, 0.985339130326871560827260802555, 1.25147645994431730424226489245, 1.27326428820631788152790280374, 1.38304850228961810857273134960, 1.58502097505968508229793693947, 1.91237660527723420637170265497, 1.96816727010849997332174532151, 1.97577303415852448482727393191, 2.00516954240082135049054479169, 2.22236277286238987950083540960, 2.37196197095986004348258665408, 2.56494558252329074986208234716, 2.59944912669301067306853146054, 2.64664613637685769588946845608, 2.72447476531906859358312338592, 2.75905827673759531140957856664, 3.03897931150445592527912206871, 3.17242810961068562627269376658, 3.22832075588624545185506941837, 3.23603288392117590536855767358, 3.39417595980536809425675077108
Plot not available for L-functions of degree greater than 10.
