| L(s) = 1 | + 16·17-s + 8·25-s + 48·41-s − 56·49-s − 16·73-s + 48·89-s − 48·97-s − 48·113-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 24·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | + 3.88·17-s + 8/5·25-s + 7.49·41-s − 8·49-s − 1.87·73-s + 5.08·89-s − 4.87·97-s − 4.51·113-s − 0.363·121-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 + 1.84·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{16} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{16} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.105862238\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.105862238\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( ( 1 + T^{2} )^{4} \) |
| good | 5 | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{4} \) |
| 7 | \( ( 1 + p T^{2} )^{8} \) |
| 13 | \( ( 1 - 12 T^{2} - 10 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 - 4 T + 14 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 19 | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{4} \) |
| 23 | \( ( 1 + 52 T^{2} + 1350 T^{4} + 52 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 - 76 T^{2} + 2742 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 + 36 T^{2} + 710 T^{4} + 36 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 - 10 T + p T^{2} )^{4}( 1 + 10 T + p T^{2} )^{4} \) |
| 41 | \( ( 1 - 12 T + 94 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{4} \) |
| 43 | \( ( 1 - 92 T^{2} + 4278 T^{4} - 92 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 + 148 T^{2} + 9510 T^{4} + 148 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - 100 T^{2} + 6582 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 - 12 T^{2} + 854 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 76 T^{2} + 5430 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 - 44 T^{2} + 3318 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 + 116 T^{2} + 9990 T^{4} + 116 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 + 110 T^{2} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{4} \) |
| 89 | \( ( 1 - 12 T + 118 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{4} \) |
| 97 | \( ( 1 + 6 T + p T^{2} )^{8} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.24852564257003489086148419126, −3.20985639444728761839697216299, −3.15271171080577199070362013007, −2.92291153501637029311076145196, −2.90357023753256584848419572247, −2.74760938380184135384393560620, −2.74434471142083341202838692865, −2.43902370927826750827609045266, −2.30276471430301992226242481025, −2.23829852114862497712439520946, −2.21549379535396436422979422779, −2.19279756130806992644253447017, −1.86410942490445144415504179899, −1.77932992077777203492731166050, −1.51098824481678950552410830286, −1.46632741219207521929730697544, −1.17095324678167537216305862055, −1.16945701638543239333726642659, −1.16196879497173546643848460791, −1.05500268496502909291381261530, −1.01386144313594253702538003943, −0.845901959536030432557539034991, −0.37352446755921169758127113052, −0.27412587009344105598489200721, −0.18213661428705363940380350285,
0.18213661428705363940380350285, 0.27412587009344105598489200721, 0.37352446755921169758127113052, 0.845901959536030432557539034991, 1.01386144313594253702538003943, 1.05500268496502909291381261530, 1.16196879497173546643848460791, 1.16945701638543239333726642659, 1.17095324678167537216305862055, 1.46632741219207521929730697544, 1.51098824481678950552410830286, 1.77932992077777203492731166050, 1.86410942490445144415504179899, 2.19279756130806992644253447017, 2.21549379535396436422979422779, 2.23829852114862497712439520946, 2.30276471430301992226242481025, 2.43902370927826750827609045266, 2.74434471142083341202838692865, 2.74760938380184135384393560620, 2.90357023753256584848419572247, 2.92291153501637029311076145196, 3.15271171080577199070362013007, 3.20985639444728761839697216299, 3.24852564257003489086148419126
Plot not available for L-functions of degree greater than 10.
