| L(s) = 1 | − 2·2-s − 4·3-s + 2·4-s + 8·6-s + 8·9-s − 8·11-s − 8·12-s − 8·17-s − 16·18-s + 16·22-s − 12·27-s − 8·32-s + 32·33-s + 16·34-s + 16·36-s − 8·41-s + 28·43-s − 16·44-s + 32·51-s + 24·54-s + 16·64-s − 64·66-s − 28·67-s − 16·68-s + 16·73-s + 24·81-s + 16·82-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 2.30·3-s + 4-s + 3.26·6-s + 8/3·9-s − 2.41·11-s − 2.30·12-s − 1.94·17-s − 3.77·18-s + 3.41·22-s − 2.30·27-s − 1.41·32-s + 5.57·33-s + 2.74·34-s + 8/3·36-s − 1.24·41-s + 4.26·43-s − 2.41·44-s + 4.48·51-s + 3.26·54-s + 2·64-s − 7.87·66-s − 3.42·67-s − 1.94·68-s + 1.87·73-s + 8/3·81-s + 1.76·82-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.03848037373\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.03848037373\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p T + p T^{2} - p^{2} T^{4} + p^{3} T^{6} + p^{4} T^{7} + p^{4} T^{8} \) |
| 5 | \( 1 + 6 p T^{4} + p^{4} T^{8} \) |
| good | 3 | \( ( 1 + 2 T + 2 T^{2} + 2 T^{3} - 2 T^{4} + 2 p T^{5} + 2 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 7 | \( 1 - 24 T^{4} + 3326 T^{8} - 24 p^{4} T^{12} + p^{8} T^{16} \) |
| 11 | \( ( 1 + 2 T + 18 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{4} \) |
| 13 | \( 1 - 4 T^{4} - 24794 T^{8} - 4 p^{4} T^{12} + p^{8} T^{16} \) |
| 17 | \( ( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{4} \) |
| 19 | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{4} \) |
| 23 | \( 1 - 984 T^{4} + 581246 T^{8} - 984 p^{4} T^{12} + p^{8} T^{16} \) |
| 29 | \( ( 1 + 76 T^{2} + 2806 T^{4} + 76 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 - 24 T^{2} + 1566 T^{4} - 24 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( 1 + 2876 T^{4} + 4505446 T^{8} + 2876 p^{4} T^{12} + p^{8} T^{16} \) |
| 41 | \( ( 1 + 2 T + 38 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{4} \) |
| 43 | \( ( 1 - 14 T + 98 T^{2} - 910 T^{3} + 7966 T^{4} - 910 p T^{5} + 98 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 47 | \( 1 - 3544 T^{4} + 6827326 T^{8} - 3544 p^{4} T^{12} + p^{8} T^{16} \) |
| 53 | \( 1 + 7356 T^{4} + 26359526 T^{8} + 7356 p^{4} T^{12} + p^{8} T^{16} \) |
| 59 | \( ( 1 - 164 T^{2} + 12406 T^{4} - 164 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 144 T^{2} + 10206 T^{4} - 144 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 + 14 T + 98 T^{2} + 1246 T^{3} + 15358 T^{4} + 1246 p T^{5} + 98 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 - 104 T^{2} + 11166 T^{4} - 104 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 - 8 T + 32 T^{2} + 72 T^{3} - 6562 T^{4} + 72 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 + 156 T^{2} + 13446 T^{4} + 156 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( ( 1 + 22 T + 242 T^{2} + 3102 T^{3} + 36398 T^{4} + 3102 p T^{5} + 242 p^{2} T^{6} + 22 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 - 308 T^{2} + 39238 T^{4} - 308 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 - 8 T + 32 T^{2} - 760 T^{3} + 18046 T^{4} - 760 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
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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
−7.987190908828908892803373876433, −7.81073265124145177978484073132, −7.48333107972619529959910072324, −7.44565328305927917049725714879, −7.39535373805772663047943239386, −7.36348840895819346221251807431, −6.88986761452075623932627357931, −6.75377824996975443850607170105, −6.52421587298763225381509625730, −6.17863305045664836009249897795, −5.90558389948330060439191700396, −5.89554168775237004951217578524, −5.72626834946188853059300205419, −5.65463646727497286343915366680, −5.04691926888097468421191304043, −5.02815510561311717267520628090, −4.78801037182185899462013514513, −4.63270301901924420749172501689, −4.32096800543823176313844239520, −3.89312045284480504322586550705, −3.73976913195957760440073579450, −3.09210878135863900598836812375, −2.65222453251457968239482593243, −2.39783784686601135498057015225, −1.77368710749098304089451956981,
1.77368710749098304089451956981, 2.39783784686601135498057015225, 2.65222453251457968239482593243, 3.09210878135863900598836812375, 3.73976913195957760440073579450, 3.89312045284480504322586550705, 4.32096800543823176313844239520, 4.63270301901924420749172501689, 4.78801037182185899462013514513, 5.02815510561311717267520628090, 5.04691926888097468421191304043, 5.65463646727497286343915366680, 5.72626834946188853059300205419, 5.89554168775237004951217578524, 5.90558389948330060439191700396, 6.17863305045664836009249897795, 6.52421587298763225381509625730, 6.75377824996975443850607170105, 6.88986761452075623932627357931, 7.36348840895819346221251807431, 7.39535373805772663047943239386, 7.44565328305927917049725714879, 7.48333107972619529959910072324, 7.81073265124145177978484073132, 7.987190908828908892803373876433
Plot not available for L-functions of degree greater than 10.
