| L(s) = 1 | − 4·5-s − 6·9-s − 32·13-s + 52·17-s − 48·25-s − 8·29-s + 80·37-s − 68·41-s + 24·45-s + 16·53-s + 264·61-s + 128·65-s + 272·73-s + 27·81-s − 208·85-s + 172·89-s − 320·97-s + 436·101-s − 88·109-s + 424·113-s + 192·117-s + 464·121-s + 228·125-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 4/5·5-s − 2/3·9-s − 2.46·13-s + 3.05·17-s − 1.91·25-s − 0.275·29-s + 2.16·37-s − 1.65·41-s + 8/15·45-s + 0.301·53-s + 4.32·61-s + 1.96·65-s + 3.72·73-s + 1/3·81-s − 2.44·85-s + 1.93·89-s − 3.29·97-s + 4.31·101-s − 0.807·109-s + 3.75·113-s + 1.64·117-s + 3.83·121-s + 1.82·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(5.364433053\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.364433053\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $D_{4}$ | \( ( 1 + 2 T + 6 p T^{2} + 2 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 - 464 T^{2} + 83022 T^{4} - 464 p^{4} T^{6} + p^{8} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 16 T + 318 T^{2} + 16 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 17 | $D_{4}$ | \( ( 1 - 26 T + 558 T^{2} - 26 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 1124 T^{2} + 554982 T^{4} - 1124 p^{4} T^{6} + p^{8} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 1280 T^{2} + 866382 T^{4} - 1280 p^{4} T^{6} + p^{8} T^{8} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p^{2} T^{2} )^{4} \) |
| 31 | $D_4\times C_2$ | \( 1 - 1444 T^{2} + 1594182 T^{4} - 1444 p^{4} T^{6} + p^{8} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 - 40 T + 2382 T^{2} - 40 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 + 34 T + 102 T^{2} + 34 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 2276 T^{2} + 2627622 T^{4} - 2276 p^{4} T^{6} + p^{8} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 7156 T^{2} + 21968742 T^{4} - 7156 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 - 8 T + 3534 T^{2} - 8 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 5204 T^{2} + 17832582 T^{4} - 5204 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - 132 T + 8774 T^{2} - 132 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 12244 T^{2} + 72446790 T^{4} - 12244 p^{4} T^{6} + p^{8} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 12704 T^{2} + 90771342 T^{4} - 12704 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 136 T + 14526 T^{2} - 136 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 - 11764 T^{2} + 106099302 T^{4} - 11764 p^{4} T^{6} + p^{8} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 13124 T^{2} + 100043430 T^{4} - 13124 p^{4} T^{6} + p^{8} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 86 T + 11622 T^{2} - 86 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 + 160 T + 24462 T^{2} + 160 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.24666005125534953115223614269, −5.81864141987371806234379322813, −5.74077286955525622593071484470, −5.53235187102480080306792410962, −5.48986672080343787235859295435, −4.98270140691602400346375030752, −4.93125317905845821820078620473, −4.85731647006165248382884759915, −4.73421578266227239571332649285, −4.00906684809652610749078244715, −3.96537874296615215659723726840, −3.87338681561528541604838307060, −3.74239688550247573896395756395, −3.30423873242890170150474385739, −3.11450493761426200643419194707, −2.88021281751218388185183933981, −2.79287603186568256330264325231, −2.20544564036786792349962734169, −2.11968399458247026660151130961, −1.87471093353470660055538753674, −1.78520858896390217917524749458, −0.854620506712297607983094383055, −0.841891824488564857182415727780, −0.48659485308720544301425632875, −0.46071544798692331345826567640,
0.46071544798692331345826567640, 0.48659485308720544301425632875, 0.841891824488564857182415727780, 0.854620506712297607983094383055, 1.78520858896390217917524749458, 1.87471093353470660055538753674, 2.11968399458247026660151130961, 2.20544564036786792349962734169, 2.79287603186568256330264325231, 2.88021281751218388185183933981, 3.11450493761426200643419194707, 3.30423873242890170150474385739, 3.74239688550247573896395756395, 3.87338681561528541604838307060, 3.96537874296615215659723726840, 4.00906684809652610749078244715, 4.73421578266227239571332649285, 4.85731647006165248382884759915, 4.93125317905845821820078620473, 4.98270140691602400346375030752, 5.48986672080343787235859295435, 5.53235187102480080306792410962, 5.74077286955525622593071484470, 5.81864141987371806234379322813, 6.24666005125534953115223614269
