| L(s) = 1 | + 3·3-s + 2·4-s + 5-s + 7-s + 3·9-s + 6·12-s + 4·13-s + 3·15-s − 2·17-s + 6·19-s + 2·20-s + 3·21-s − 20·23-s + 5·25-s + 2·28-s − 10·29-s − 31-s + 35-s + 6·36-s + 5·37-s + 12·39-s + 2·41-s − 32·43-s + 3·45-s − 8·47-s − 6·51-s + 8·52-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 4-s + 0.447·5-s + 0.377·7-s + 9-s + 1.73·12-s + 1.10·13-s + 0.774·15-s − 0.485·17-s + 1.37·19-s + 0.447·20-s + 0.654·21-s − 4.17·23-s + 25-s + 0.377·28-s − 1.85·29-s − 0.179·31-s + 0.169·35-s + 36-s + 0.821·37-s + 1.92·39-s + 0.312·41-s − 4.87·43-s + 0.447·45-s − 1.16·47-s − 0.840·51-s + 1.10·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.331774610\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.331774610\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 11 | | \( 1 \) |
| good | 2 | $C_4\times C_2$ | \( 1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8} \) |
| 3 | $C_4\times C_2$ | \( 1 - p T + 2 p T^{2} - p^{2} T^{3} + p^{2} T^{4} - p^{3} T^{5} + 2 p^{3} T^{6} - p^{4} T^{7} + p^{4} T^{8} \) |
| 5 | $C_4\times C_2$ | \( 1 - T - 4 T^{2} + 9 T^{3} + 11 T^{4} + 9 p T^{5} - 4 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_4\times C_2$ | \( 1 - 4 T + 3 T^{2} + 40 T^{3} - 199 T^{4} + 40 p T^{5} + 3 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_4\times C_2$ | \( 1 + 2 T - 13 T^{2} - 60 T^{3} + 101 T^{4} - 60 p T^{5} - 13 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_4\times C_2$ | \( 1 - 6 T + 17 T^{2} + 12 T^{3} - 395 T^{4} + 12 p T^{5} + 17 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{4} \) |
| 29 | $C_4\times C_2$ | \( 1 + 10 T + 71 T^{2} + 420 T^{3} + 2141 T^{4} + 420 p T^{5} + 71 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_4\times C_2$ | \( 1 + T - 30 T^{2} - 61 T^{3} + 869 T^{4} - 61 p T^{5} - 30 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_4\times C_2$ | \( 1 - 5 T - 12 T^{2} + 245 T^{3} - 781 T^{4} + 245 p T^{5} - 12 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_4\times C_2$ | \( 1 - 2 T - 37 T^{2} + 156 T^{3} + 1205 T^{4} + 156 p T^{5} - 37 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) |
| 47 | $C_4\times C_2$ | \( 1 + 8 T + 17 T^{2} - 240 T^{3} - 2719 T^{4} - 240 p T^{5} + 17 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_4\times C_2$ | \( 1 - 6 T - 17 T^{2} + 420 T^{3} - 1619 T^{4} + 420 p T^{5} - 17 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_4\times C_2$ | \( 1 + 3 T - 50 T^{2} - 327 T^{3} + 1969 T^{4} - 327 p T^{5} - 50 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_4\times C_2$ | \( 1 - 2 T - 57 T^{2} + 236 T^{3} + 3005 T^{4} + 236 p T^{5} - 57 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{4} \) |
| 71 | $C_4\times C_2$ | \( 1 + T - 70 T^{2} - 141 T^{3} + 4829 T^{4} - 141 p T^{5} - 70 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_4\times C_2$ | \( 1 + 10 T + 27 T^{2} - 460 T^{3} - 6571 T^{4} - 460 p T^{5} + 27 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_4\times C_2$ | \( 1 + 6 T - 43 T^{2} - 732 T^{3} - 995 T^{4} - 732 p T^{5} - 43 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_4\times C_2$ | \( 1 + 12 T + 61 T^{2} - 264 T^{3} - 8231 T^{4} - 264 p T^{5} + 61 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2$ | \( ( 1 + 15 T + p T^{2} )^{4} \) |
| 97 | $C_4\times C_2$ | \( 1 - 5 T - 72 T^{2} + 845 T^{3} + 2759 T^{4} + 845 p T^{5} - 72 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−7.23650483107099833224299975203, −7.18583560045705690708390309795, −6.95619799680519449721577433088, −6.71843770074205831825508880950, −6.45746843615157233125644712578, −6.08442812907609159812148395059, −5.83414094097184979341264991970, −5.80029400882931160556507219633, −5.71489902108833425034507148382, −5.29667401117005760025988128442, −4.91111853859233805660516725604, −4.47588821917921845466291307550, −4.42618131772671582762030967788, −4.24545614410316390284225995878, −3.69280650178484932255531239105, −3.52164880763805627811646996152, −3.34293613773245273288453843195, −2.99348622704762871006537909329, −2.95038797261019570639302863580, −2.37437655398428988609208952775, −2.11043495212053107590904379756, −1.76716494670071560451818519596, −1.59698148138712819478518166525, −1.56563894758485742161374946476, −0.36429114385404090000695432094,
0.36429114385404090000695432094, 1.56563894758485742161374946476, 1.59698148138712819478518166525, 1.76716494670071560451818519596, 2.11043495212053107590904379756, 2.37437655398428988609208952775, 2.95038797261019570639302863580, 2.99348622704762871006537909329, 3.34293613773245273288453843195, 3.52164880763805627811646996152, 3.69280650178484932255531239105, 4.24545614410316390284225995878, 4.42618131772671582762030967788, 4.47588821917921845466291307550, 4.91111853859233805660516725604, 5.29667401117005760025988128442, 5.71489902108833425034507148382, 5.80029400882931160556507219633, 5.83414094097184979341264991970, 6.08442812907609159812148395059, 6.45746843615157233125644712578, 6.71843770074205831825508880950, 6.95619799680519449721577433088, 7.18583560045705690708390309795, 7.23650483107099833224299975203
