| L(s) = 1 | − 2·4-s − 4·5-s + 3·16-s + 16·19-s + 8·20-s + 8·25-s − 8·29-s − 16·31-s + 24·41-s − 16·59-s − 24·61-s − 4·64-s − 24·71-s − 32·76-s − 8·79-s − 12·80-s − 18·81-s − 40·89-s − 64·95-s − 16·100-s − 24·101-s − 8·109-s + 16·116-s + 4·121-s + 32·124-s − 20·125-s + 127-s + ⋯ |
| L(s) = 1 | − 4-s − 1.78·5-s + 3/4·16-s + 3.67·19-s + 1.78·20-s + 8/5·25-s − 1.48·29-s − 2.87·31-s + 3.74·41-s − 2.08·59-s − 3.07·61-s − 1/2·64-s − 2.84·71-s − 3.67·76-s − 0.900·79-s − 1.34·80-s − 2·81-s − 4.23·89-s − 6.56·95-s − 8/5·100-s − 2.38·101-s − 0.766·109-s + 1.48·116-s + 4/11·121-s + 2.87·124-s − 1.78·125-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1397390913\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1397390913\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 32 T^{2} + 498 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 - 8 T + 48 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 36 T^{2} + 998 T^{4} - 36 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 4 T + 38 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 - 12 T + 94 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 92 T^{2} + 4278 T^{4} - 92 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 108 T^{2} + 5798 T^{4} - 108 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 92 T^{2} + 4278 T^{4} - 92 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 8 T + 128 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 + 12 T + 152 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $D_{4}$ | \( ( 1 + 12 T + 154 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 236 T^{2} + 24198 T^{4} - 236 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 4 T + 138 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 160 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{4} \) |
| 97 | $D_4\times C_2$ | \( 1 - 124 T^{2} + 8838 T^{4} - 124 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−7.906338874764839013728995370057, −7.53297166757971703566699406231, −7.51815889447922875993865195530, −7.40291975097973064529502854162, −7.19236681637688387315842701221, −6.98123750148877122484900315025, −6.48151124622389409407898717838, −5.95584051141790940636650169868, −5.82524168539910398593475117643, −5.60063092209570976289119478499, −5.55708497476079867512183240825, −5.13906010295543071750474090957, −4.94241521212648658785106116413, −4.32038634891721982666113220059, −4.23042896156416899809120109843, −4.12873622388879084565059835855, −3.98861477662556925968682259245, −3.26684341551782606088953771249, −3.18489074925452591269683327485, −2.96662214029900634387563728569, −2.81336266714014496872473979834, −1.84470633503726856444628060390, −1.28296491791730939813381909729, −1.26772411955896465249025104503, −0.14652777064952203964227770508,
0.14652777064952203964227770508, 1.26772411955896465249025104503, 1.28296491791730939813381909729, 1.84470633503726856444628060390, 2.81336266714014496872473979834, 2.96662214029900634387563728569, 3.18489074925452591269683327485, 3.26684341551782606088953771249, 3.98861477662556925968682259245, 4.12873622388879084565059835855, 4.23042896156416899809120109843, 4.32038634891721982666113220059, 4.94241521212648658785106116413, 5.13906010295543071750474090957, 5.55708497476079867512183240825, 5.60063092209570976289119478499, 5.82524168539910398593475117643, 5.95584051141790940636650169868, 6.48151124622389409407898717838, 6.98123750148877122484900315025, 7.19236681637688387315842701221, 7.40291975097973064529502854162, 7.51815889447922875993865195530, 7.53297166757971703566699406231, 7.906338874764839013728995370057
