| L(s) = 1 | − 4·4-s − 12·5-s + 4·16-s − 24·17-s − 24·19-s + 48·20-s − 60·23-s + 78·25-s + 68·31-s + 240·47-s − 8·49-s + 408·53-s + 196·61-s + 16·64-s + 96·68-s + 96·76-s − 180·79-s − 48·80-s − 108·83-s + 288·85-s + 240·92-s + 288·95-s − 312·100-s − 288·107-s − 152·109-s − 48·113-s + 720·115-s + ⋯ |
| L(s) = 1 | − 4-s − 2.39·5-s + 1/4·16-s − 1.41·17-s − 1.26·19-s + 12/5·20-s − 2.60·23-s + 3.11·25-s + 2.19·31-s + 5.10·47-s − 0.163·49-s + 7.69·53-s + 3.21·61-s + 1/4·64-s + 1.41·68-s + 1.26·76-s − 2.27·79-s − 3/5·80-s − 1.30·83-s + 3.38·85-s + 2.60·92-s + 3.03·95-s − 3.11·100-s − 2.69·107-s − 1.39·109-s − 0.424·113-s + 6.26·115-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{32} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{32} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.9302333948\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9302333948\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 3 | \( 1 \) |
| 5 | \( 1 + 12 T + 66 T^{2} + 336 T^{3} + 1859 T^{4} + 336 p^{2} T^{5} + 66 p^{4} T^{6} + 12 p^{6} T^{7} + p^{8} T^{8} \) |
| good | 7 | \( 1 + 8 T^{2} + 3958 T^{4} - 69568 T^{6} + 9244771 T^{8} - 69568 p^{4} T^{10} + 3958 p^{8} T^{12} + 8 p^{12} T^{14} + p^{16} T^{16} \) |
| 11 | \( 1 + 96 T^{2} - 170 p^{2} T^{4} + 48384 T^{6} + 537915459 T^{8} + 48384 p^{4} T^{10} - 170 p^{10} T^{12} + 96 p^{12} T^{14} + p^{16} T^{16} \) |
| 13 | \( 1 + 352 T^{2} + 52006 T^{4} + 5201152 T^{6} + 814769539 T^{8} + 5201152 p^{4} T^{10} + 52006 p^{8} T^{12} + 352 p^{12} T^{14} + p^{16} T^{16} \) |
| 17 | \( ( 1 + 6 T + 489 T^{2} + 6 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 19 | \( ( 1 + 6 T + 713 T^{2} + 6 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 23 | \( ( 1 + 30 T + 9 T^{2} - 5010 T^{3} + 37940 T^{4} - 5010 p^{2} T^{5} + 9 p^{4} T^{6} + 30 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 29 | \( 1 + 1096 T^{2} - 41258 T^{4} - 188608448 T^{6} + 193897974019 T^{8} - 188608448 p^{4} T^{10} - 41258 p^{8} T^{12} + 1096 p^{12} T^{14} + p^{16} T^{16} \) |
| 31 | \( ( 1 - 34 T - 605 T^{2} + 5474 T^{3} + 1066684 T^{4} + 5474 p^{2} T^{5} - 605 p^{4} T^{6} - 34 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 37 | \( ( 1 - 68 T^{2} - 1268634 T^{4} - 68 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 41 | \( 1 + 5352 T^{2} + 16236406 T^{4} + 36157983552 T^{6} + 65020254579939 T^{8} + 36157983552 p^{4} T^{10} + 16236406 p^{8} T^{12} + 5352 p^{12} T^{14} + p^{16} T^{16} \) |
| 43 | \( 1 + 5672 T^{2} + 17968534 T^{4} + 41776821056 T^{6} + 80050185474115 T^{8} + 41776821056 p^{4} T^{10} + 17968534 p^{8} T^{12} + 5672 p^{12} T^{14} + p^{16} T^{16} \) |
| 47 | \( ( 1 - 120 T + 6774 T^{2} - 384960 T^{3} + 21466595 T^{4} - 384960 p^{2} T^{5} + 6774 p^{4} T^{6} - 120 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 53 | \( ( 1 - 102 T + 8057 T^{2} - 102 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 59 | \( 1 + 10848 T^{2} + 64062694 T^{4} + 318732551424 T^{6} + 1301928798359235 T^{8} + 318732551424 p^{4} T^{10} + 64062694 p^{8} T^{12} + 10848 p^{12} T^{14} + p^{16} T^{16} \) |
| 61 | \( ( 1 - 98 T + 913 T^{2} - 122402 T^{3} + 25951156 T^{4} - 122402 p^{2} T^{5} + 913 p^{4} T^{6} - 98 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 67 | \( 1 - 6508 T^{2} + 10043914 T^{4} + 52012534736 T^{6} - 215603753039885 T^{8} + 52012534736 p^{4} T^{10} + 10043914 p^{8} T^{12} - 6508 p^{12} T^{14} + p^{16} T^{16} \) |
| 71 | \( ( 1 - 4288 T^{2} + 45932730 T^{4} - 4288 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 73 | \( ( 1 - 13280 T^{2} + 84777594 T^{4} - 13280 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 79 | \( ( 1 + 90 T + 1531 T^{2} - 532170 T^{3} - 46350420 T^{4} - 532170 p^{2} T^{5} + 1531 p^{4} T^{6} + 90 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 83 | \( ( 1 + 54 T - 4863 T^{2} - 323946 T^{3} - 7033804 T^{4} - 323946 p^{2} T^{5} - 4863 p^{4} T^{6} + 54 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 89 | \( ( 1 - 8136 T^{2} + 91108874 T^{4} - 8136 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 97 | \( 1 + 19172 T^{2} + 99652426 T^{4} + 1741864314512 T^{6} + 31399709763290899 T^{8} + 1741864314512 p^{4} T^{10} + 99652426 p^{8} T^{12} + 19172 p^{12} T^{14} + p^{16} T^{16} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.10804013910753723116328626251, −4.04741587646894709786066585647, −4.02335260674913528056436148356, −4.00927154651713340206162425555, −3.96991360697020507313288918671, −3.77756894658837106240983324333, −3.62847622748312572550362252916, −3.03548737831584495105297620840, −2.95164571100874786191989904165, −2.94399051906611577100944918549, −2.81393725788715210346190998928, −2.76688406007992074523015923868, −2.55523594474710831199619177002, −2.27139735458653082975685804872, −2.09138761141746423621700220638, −2.03140740518457969547061455779, −1.99201252381411382649445591788, −1.81074646093997913882376712388, −1.12439497969789112115104229166, −1.07220884815496910694421035658, −0.932099603229460782758466169489, −0.78493788701566893907115934662, −0.59573891358505706121519967929, −0.28349056199335134225424521840, −0.17087934155385081822813630987,
0.17087934155385081822813630987, 0.28349056199335134225424521840, 0.59573891358505706121519967929, 0.78493788701566893907115934662, 0.932099603229460782758466169489, 1.07220884815496910694421035658, 1.12439497969789112115104229166, 1.81074646093997913882376712388, 1.99201252381411382649445591788, 2.03140740518457969547061455779, 2.09138761141746423621700220638, 2.27139735458653082975685804872, 2.55523594474710831199619177002, 2.76688406007992074523015923868, 2.81393725788715210346190998928, 2.94399051906611577100944918549, 2.95164571100874786191989904165, 3.03548737831584495105297620840, 3.62847622748312572550362252916, 3.77756894658837106240983324333, 3.96991360697020507313288918671, 4.00927154651713340206162425555, 4.02335260674913528056436148356, 4.04741587646894709786066585647, 4.10804013910753723116328626251
Plot not available for L-functions of degree greater than 10.
