| L(s) = 1 | − 4·9-s + 24·17-s − 20·25-s + 24·41-s + 28·49-s − 8·73-s + 22·81-s − 72·89-s + 60·97-s + 108·113-s + 14·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 96·153-s + 157-s + 163-s + 167-s − 52·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | − 4/3·9-s + 5.82·17-s − 4·25-s + 3.74·41-s + 4·49-s − 0.936·73-s + 22/9·81-s − 7.63·89-s + 6.09·97-s + 10.1·113-s + 1.27·121-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 − 7.76·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 4·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{96} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{96} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(11.21395246\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.21395246\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 - 14 T^{2} + 75 T^{4} + 644 T^{6} - 18091 T^{8} + 644 p^{2} T^{10} + 75 p^{4} T^{12} - 14 p^{6} T^{14} + p^{8} T^{16} \) |
| good | 3 | \( ( 1 + 2 T^{2} - 5 T^{4} - 28 T^{6} - 11 T^{8} - 28 p^{2} T^{10} - 5 p^{4} T^{12} + 2 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 5 | \( ( 1 - p T + 3 p T^{2} - p^{2} T^{3} + p^{2} T^{4} )^{4}( 1 + p T + 3 p T^{2} + p^{2} T^{3} + p^{2} T^{4} )^{4} \) |
| 7 | \( ( 1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8} )^{4} \) |
| 13 | \( ( 1 + p T^{2} + p^{2} T^{4} + p^{3} T^{6} + p^{4} T^{8} )^{4} \) |
| 17 | \( ( 1 - 6 T + 19 T^{2} - 12 T^{3} - 251 T^{4} - 12 p T^{5} + 19 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{4} \) |
| 19 | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{4}( 1 + 34 T^{2} + 795 T^{4} + 14756 T^{6} + 214709 T^{8} + 14756 p^{2} T^{10} + 795 p^{4} T^{12} + 34 p^{6} T^{14} + p^{8} T^{16} ) \) |
| 23 | \( ( 1 + p T^{2} )^{16} \) |
| 29 | \( ( 1 + p T^{2} + p^{2} T^{4} + p^{3} T^{6} + p^{4} T^{8} )^{4} \) |
| 31 | \( ( 1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8} )^{4} \) |
| 37 | \( ( 1 + p T^{2} + p^{2} T^{4} + p^{3} T^{6} + p^{4} T^{8} )^{4} \) |
| 41 | \( ( 1 - 6 T - 5 T^{2} + 276 T^{3} - 1451 T^{4} + 276 p T^{5} - 5 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{4} \) |
| 43 | \( ( 1 - 14 T^{2} - 1653 T^{4} + 49028 T^{6} + 2370005 T^{8} + 49028 p^{2} T^{10} - 1653 p^{4} T^{12} - 14 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 47 | \( ( 1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8} )^{4} \) |
| 53 | \( ( 1 + p T^{2} + p^{2} T^{4} + p^{3} T^{6} + p^{4} T^{8} )^{4} \) |
| 59 | \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{4}( 1 + 82 T^{2} + 3243 T^{4} - 19516 T^{6} - 12889195 T^{8} - 19516 p^{2} T^{10} + 3243 p^{4} T^{12} + 82 p^{6} T^{14} + p^{8} T^{16} ) \) |
| 61 | \( ( 1 + p T^{2} + p^{2} T^{4} + p^{3} T^{6} + p^{4} T^{8} )^{4} \) |
| 67 | \( ( 1 - 62 T^{2} - 645 T^{4} + 318308 T^{6} - 16839691 T^{8} + 318308 p^{2} T^{10} - 645 p^{4} T^{12} - 62 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 71 | \( ( 1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8} )^{4} \) |
| 73 | \( ( 1 + 2 T - 69 T^{2} - 284 T^{3} + 4469 T^{4} - 284 p T^{5} - 69 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{4} \) |
| 79 | \( ( 1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8} )^{4} \) |
| 83 | \( ( 1 + 158 T^{2} + p^{2} T^{4} )^{4}( 1 - 158 T^{2} + 18075 T^{4} - 1767388 T^{6} + 154728629 T^{8} - 1767388 p^{2} T^{10} + 18075 p^{4} T^{12} - 158 p^{6} T^{14} + p^{8} T^{16} ) \) |
| 89 | \( ( 1 + 18 T + 235 T^{2} + 2628 T^{3} + 26389 T^{4} + 2628 p T^{5} + 235 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} )^{4} \) |
| 97 | \( ( 1 - 10 T + p T^{2} )^{8}( 1 + 10 T + 3 T^{2} - 940 T^{3} - 9691 T^{4} - 940 p T^{5} + 3 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
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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
−2.84762017626817316656898731767, −2.79917119792363199882366083526, −2.52373506804093128702283812892, −2.51555973650438898590491190820, −2.40061833493722135561365323198, −2.37173455064092744305873050800, −2.27707034476471018965670988006, −2.15713383250354655914742183378, −2.13297532527041161196204801000, −2.13070108179221010863417472521, −1.89502979543863693023211249017, −1.81134744388610302689277927395, −1.75678854797318774708224405510, −1.71823004770851093921198670666, −1.54078094195192665570188818000, −1.30487883772282110528240025550, −1.23208237037704920447254197833, −1.08870911095629259410383174644, −1.05563560105422264859844092152, −1.00200726372320572326732205392, −0.930405850206352158870215634051, −0.64436947775834886002167794390, −0.49937030872748843437906586378, −0.49382649743816061543380811713, −0.19906754559800217465764310062,
0.19906754559800217465764310062, 0.49382649743816061543380811713, 0.49937030872748843437906586378, 0.64436947775834886002167794390, 0.930405850206352158870215634051, 1.00200726372320572326732205392, 1.05563560105422264859844092152, 1.08870911095629259410383174644, 1.23208237037704920447254197833, 1.30487883772282110528240025550, 1.54078094195192665570188818000, 1.71823004770851093921198670666, 1.75678854797318774708224405510, 1.81134744388610302689277927395, 1.89502979543863693023211249017, 2.13070108179221010863417472521, 2.13297532527041161196204801000, 2.15713383250354655914742183378, 2.27707034476471018965670988006, 2.37173455064092744305873050800, 2.40061833493722135561365323198, 2.51555973650438898590491190820, 2.52373506804093128702283812892, 2.79917119792363199882366083526, 2.84762017626817316656898731767
Plot not available for L-functions of degree greater than 10.
