| L(s) = 1 | − 8·4-s − 40·13-s + 40·16-s + 40·19-s + 100·25-s − 40·31-s − 8·37-s + 40·43-s + 28·49-s + 320·52-s − 88·61-s − 160·64-s + 240·67-s − 288·73-s − 320·76-s − 424·79-s + 600·97-s − 800·100-s − 320·103-s − 216·109-s + 616·121-s + 320·124-s + 127-s + 131-s + 137-s + 139-s + 64·148-s + ⋯ |
| L(s) = 1 | − 2·4-s − 3.07·13-s + 5/2·16-s + 2.10·19-s + 4·25-s − 1.29·31-s − 0.216·37-s + 0.930·43-s + 4/7·49-s + 6.15·52-s − 1.44·61-s − 5/2·64-s + 3.58·67-s − 3.94·73-s − 4.21·76-s − 5.36·79-s + 6.18·97-s − 8·100-s − 3.10·103-s − 1.98·109-s + 5.09·121-s + 2.58·124-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.432·148-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{24} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{24} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.3171191468\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3171191468\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + p T^{2} )^{4} \) |
| 3 | \( 1 \) |
| 7 | \( ( 1 - p T^{2} )^{4} \) |
| good | 5 | \( 1 - 4 p^{2} T^{2} + 4874 T^{4} - 165072 T^{6} + 4508819 T^{8} - 165072 p^{4} T^{10} + 4874 p^{8} T^{12} - 4 p^{14} T^{14} + p^{16} T^{16} \) |
| 11 | \( 1 - 56 p T^{2} + 192932 T^{4} - 3565800 p T^{6} + 5596633670 T^{8} - 3565800 p^{5} T^{10} + 192932 p^{8} T^{12} - 56 p^{13} T^{14} + p^{16} T^{16} \) |
| 13 | \( ( 1 + 20 T + 524 T^{2} + 8628 T^{3} + 127802 T^{4} + 8628 p^{2} T^{5} + 524 p^{4} T^{6} + 20 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 17 | \( 1 - 1204 T^{2} + 813818 T^{4} - 365052912 T^{6} + 122024410499 T^{8} - 365052912 p^{4} T^{10} + 813818 p^{8} T^{12} - 1204 p^{12} T^{14} + p^{16} T^{16} \) |
| 19 | \( ( 1 - 20 T + 1180 T^{2} - 21044 T^{3} + 594202 T^{4} - 21044 p^{2} T^{5} + 1180 p^{4} T^{6} - 20 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 23 | \( 1 - 2296 T^{2} + 2899868 T^{4} - 2458609608 T^{6} + 1516240068038 T^{8} - 2458609608 p^{4} T^{10} + 2899868 p^{8} T^{12} - 2296 p^{12} T^{14} + p^{16} T^{16} \) |
| 29 | \( 1 - 4360 T^{2} + 9761444 T^{4} - 14042297304 T^{6} + 14026596164294 T^{8} - 14042297304 p^{4} T^{10} + 9761444 p^{8} T^{12} - 4360 p^{12} T^{14} + p^{16} T^{16} \) |
| 31 | \( ( 1 + 20 T + 1388 T^{2} + 23028 T^{3} + 1637498 T^{4} + 23028 p^{2} T^{5} + 1388 p^{4} T^{6} + 20 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 37 | \( ( 1 + 4 T + 3826 T^{2} - 11072 T^{3} + 6661243 T^{4} - 11072 p^{2} T^{5} + 3826 p^{4} T^{6} + 4 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 41 | \( 1 - 8228 T^{2} + 35127562 T^{4} - 98029902224 T^{6} + 194099437201747 T^{8} - 98029902224 p^{4} T^{10} + 35127562 p^{8} T^{12} - 8228 p^{12} T^{14} + p^{16} T^{16} \) |
| 43 | \( ( 1 - 20 T + 2938 T^{2} - 1520 p T^{3} + 8123251 T^{4} - 1520 p^{3} T^{5} + 2938 p^{4} T^{6} - 20 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 47 | \( 1 - 2948 T^{2} + 18557962 T^{4} - 39341933840 T^{6} + 134156483159059 T^{8} - 39341933840 p^{4} T^{10} + 18557962 p^{8} T^{12} - 2948 p^{12} T^{14} + p^{16} T^{16} \) |
| 53 | \( 1 - 12536 T^{2} + 57316060 T^{4} - 108075473480 T^{6} + 133851976324486 T^{8} - 108075473480 p^{4} T^{10} + 57316060 p^{8} T^{12} - 12536 p^{12} T^{14} + p^{16} T^{16} \) |
| 59 | \( 1 - 19300 T^{2} + 179852138 T^{4} - 1070283526800 T^{6} + 4427328327950003 T^{8} - 1070283526800 p^{4} T^{10} + 179852138 p^{8} T^{12} - 19300 p^{12} T^{14} + p^{16} T^{16} \) |
| 61 | \( ( 1 + 44 T + 6908 T^{2} + 475404 T^{3} + 26960282 T^{4} + 475404 p^{2} T^{5} + 6908 p^{4} T^{6} + 44 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 67 | \( ( 1 - 120 T + 15620 T^{2} - 937320 T^{3} + 80906214 T^{4} - 937320 p^{2} T^{5} + 15620 p^{4} T^{6} - 120 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 71 | \( 1 - 30280 T^{2} + 422649116 T^{4} - 3639855519480 T^{6} + 21683770287280070 T^{8} - 3639855519480 p^{4} T^{10} + 422649116 p^{8} T^{12} - 30280 p^{12} T^{14} + p^{16} T^{16} \) |
| 73 | \( ( 1 + 144 T + 24764 T^{2} + 2064240 T^{3} + 198900294 T^{4} + 2064240 p^{2} T^{5} + 24764 p^{4} T^{6} + 144 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 79 | \( ( 1 + 212 T + 35810 T^{2} + 3817056 T^{3} + 355692731 T^{4} + 3817056 p^{2} T^{5} + 35810 p^{4} T^{6} + 212 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 83 | \( 1 - 25204 T^{2} + 392841242 T^{4} - 4131882491760 T^{6} + 32900659940774435 T^{8} - 4131882491760 p^{4} T^{10} + 392841242 p^{8} T^{12} - 25204 p^{12} T^{14} + p^{16} T^{16} \) |
| 89 | \( 1 - 38024 T^{2} + 771017500 T^{4} - 10120789220024 T^{6} + 94520833820645830 T^{8} - 10120789220024 p^{4} T^{10} + 771017500 p^{8} T^{12} - 38024 p^{12} T^{14} + p^{16} T^{16} \) |
| 97 | \( ( 1 - 300 T + 53324 T^{2} - 6171180 T^{3} + 635998170 T^{4} - 6171180 p^{2} T^{5} + 53324 p^{4} T^{6} - 300 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
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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
−4.84792121982265185485611339897, −4.77119400198231199698841960803, −4.48839336119847833002834418139, −4.37258798813333715384531961776, −4.34676465417062552500811679599, −4.09742752143146985357131842752, −3.93459507302220008632887141437, −3.78452971114848436232261318565, −3.64024932543029157644023494036, −3.45508598082595501175893503340, −3.00777931419091689064592880533, −2.96844787555147698095560084697, −2.96245858749730092027193178325, −2.92362091207191860074027091262, −2.69227661645137965654041865725, −2.56891288562961892915814841662, −2.10065890830804205460037297881, −1.93719322175036305018629108613, −1.77364632074565031377704686976, −1.36824445190420871863605761703, −1.27756424294245257637846174792, −0.847775297730941222290929182547, −0.819983883543691026213721878254, −0.46529933564425594211919187127, −0.090218820717411655841853599216,
0.090218820717411655841853599216, 0.46529933564425594211919187127, 0.819983883543691026213721878254, 0.847775297730941222290929182547, 1.27756424294245257637846174792, 1.36824445190420871863605761703, 1.77364632074565031377704686976, 1.93719322175036305018629108613, 2.10065890830804205460037297881, 2.56891288562961892915814841662, 2.69227661645137965654041865725, 2.92362091207191860074027091262, 2.96245858749730092027193178325, 2.96844787555147698095560084697, 3.00777931419091689064592880533, 3.45508598082595501175893503340, 3.64024932543029157644023494036, 3.78452971114848436232261318565, 3.93459507302220008632887141437, 4.09742752143146985357131842752, 4.34676465417062552500811679599, 4.37258798813333715384531961776, 4.48839336119847833002834418139, 4.77119400198231199698841960803, 4.84792121982265185485611339897
Plot not available for L-functions of degree greater than 10.
