| L(s) = 1 | − 6·2-s + 15·4-s − 12·7-s − 18·8-s − 10·9-s + 72·14-s + 5·16-s + 60·18-s − 180·28-s − 150·36-s + 50·49-s + 216·56-s + 24·61-s + 120·63-s + 78·64-s + 48·67-s + 180·72-s + 48·79-s + 43·81-s + 24·97-s − 300·98-s + 36·101-s − 60·112-s + 30·121-s − 144·122-s − 720·126-s + 127-s + ⋯ |
| L(s) = 1 | − 4.24·2-s + 15/2·4-s − 4.53·7-s − 6.36·8-s − 3.33·9-s + 19.2·14-s + 5/4·16-s + 14.1·18-s − 34.0·28-s − 25·36-s + 50/7·49-s + 28.8·56-s + 3.07·61-s + 15.1·63-s + 39/4·64-s + 5.86·67-s + 21.2·72-s + 5.40·79-s + 43/9·81-s + 2.43·97-s − 30.3·98-s + 3.58·101-s − 5.66·112-s + 2.72·121-s − 13.0·122-s − 64.1·126-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.02240951104\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.02240951104\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - 34 T^{4} + p^{4} T^{8} \) |
| 13 | \( 1 \) |
| good | 2 | \( ( 1 + T + p T^{2} )^{4}( 1 + T - T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 3 | \( ( 1 + 5 T^{2} + 16 T^{4} + 5 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 7 | \( ( 1 - T + p T^{2} )^{4}( 1 + 4 T + p T^{2} )^{4} \) |
| 11 | \( ( 1 - 15 T^{2} + 104 T^{4} - 15 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 + 13 T^{2} - 120 T^{4} + 13 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 - 35 T^{2} + 864 T^{4} - 35 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 + 25 T^{2} + 96 T^{4} + 25 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 - 37 T^{2} + 528 T^{4} - 37 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 - 8 T^{2} + 1182 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 + 11 T^{2} - 1248 T^{4} + 11 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 - 75 T^{2} + 3944 T^{4} - 75 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 + 58 T^{2} + p^{2} T^{4} )^{2}( 1 - 58 T^{2} + 1515 T^{4} - 58 p^{2} T^{6} + p^{4} T^{8} ) \) |
| 47 | \( ( 1 - 108 T^{2} + 5990 T^{4} - 108 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - 152 T^{2} + 10638 T^{4} - 152 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( 1 - 30 T^{2} + 2113 T^{4} + 245250 T^{6} - 13778892 T^{8} + 245250 p^{2} T^{10} + 2113 p^{4} T^{12} - 30 p^{6} T^{14} + p^{8} T^{16} \) |
| 61 | \( ( 1 - 12 T + 7 T^{2} - 180 T^{3} + 6264 T^{4} - 180 p T^{5} + 7 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 - 24 T + 311 T^{2} - 2856 T^{3} + 22536 T^{4} - 2856 p T^{5} + 311 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 71 | \( 1 - 222 T^{2} + 27217 T^{4} - 2660670 T^{6} + 213100164 T^{8} - 2660670 p^{2} T^{10} + 27217 p^{4} T^{12} - 222 p^{6} T^{14} + p^{8} T^{16} \) |
| 73 | \( ( 1 - p T^{2} )^{8} \) |
| 79 | \( ( 1 - 6 T + p T^{2} )^{8} \) |
| 83 | \( ( 1 - 168 T^{2} + 18734 T^{4} - 168 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( 1 - 246 T^{2} + 30889 T^{4} - 3391110 T^{6} + 335669652 T^{8} - 3391110 p^{2} T^{10} + 30889 p^{4} T^{12} - 246 p^{6} T^{14} + p^{8} T^{16} \) |
| 97 | \( ( 1 - 12 T + 191 T^{2} - 1716 T^{3} + 15696 T^{4} - 1716 p T^{5} + 191 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
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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
−4.44125933120349088252253436893, −4.37779141437114794348585996770, −3.79860969024694299915850814700, −3.67159808892425285815519337912, −3.64230369671617509069510941821, −3.61352282132408510198120379344, −3.38285625969659846949786558759, −3.33155712632883462203033370490, −3.31479166771994521390224331639, −3.27367463396918081345381706343, −3.26818735905267628382668842723, −2.68464434894639128991461990557, −2.49535755684036579207641542211, −2.37468395220934948346871344851, −2.33880702329700851019920244773, −2.19046719838454468366153421879, −2.13453202390823992754243392688, −2.00383282773303950351973464000, −1.40143656070778313428923789867, −1.20863887016209548440570222056, −0.842864669897239003186572351751, −0.63098984543899879391330488302, −0.50865386392471231710718420733, −0.40415935440546425687616938387, −0.21839756590916806526859545790,
0.21839756590916806526859545790, 0.40415935440546425687616938387, 0.50865386392471231710718420733, 0.63098984543899879391330488302, 0.842864669897239003186572351751, 1.20863887016209548440570222056, 1.40143656070778313428923789867, 2.00383282773303950351973464000, 2.13453202390823992754243392688, 2.19046719838454468366153421879, 2.33880702329700851019920244773, 2.37468395220934948346871344851, 2.49535755684036579207641542211, 2.68464434894639128991461990557, 3.26818735905267628382668842723, 3.27367463396918081345381706343, 3.31479166771994521390224331639, 3.33155712632883462203033370490, 3.38285625969659846949786558759, 3.61352282132408510198120379344, 3.64230369671617509069510941821, 3.67159808892425285815519337912, 3.79860969024694299915850814700, 4.37779141437114794348585996770, 4.44125933120349088252253436893
Plot not available for L-functions of degree greater than 10.
