| L(s) = 1 | + 2·2-s + 3-s + 4-s + 6·5-s + 2·6-s − 10·7-s − 2·8-s + 3·9-s + 12·10-s + 9·11-s + 12-s − 8·13-s − 20·14-s + 6·15-s − 4·16-s + 6·18-s − 12·19-s + 6·20-s − 10·21-s + 18·22-s + 3·23-s − 2·24-s + 17·25-s − 16·26-s + 8·27-s − 10·28-s + 12·30-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 0.577·3-s + 1/2·4-s + 2.68·5-s + 0.816·6-s − 3.77·7-s − 0.707·8-s + 9-s + 3.79·10-s + 2.71·11-s + 0.288·12-s − 2.21·13-s − 5.34·14-s + 1.54·15-s − 16-s + 1.41·18-s − 2.75·19-s + 1.34·20-s − 2.18·21-s + 3.83·22-s + 0.625·23-s − 0.408·24-s + 17/5·25-s − 3.13·26-s + 1.53·27-s − 1.88·28-s + 2.19·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.019808274\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.019808274\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 3 | $C_2^2$ | \( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| good | 11 | $D_4\times C_2$ | \( 1 - 9 T + 53 T^{2} - 234 T^{3} + 852 T^{4} - 234 p T^{5} + 53 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 17 | $C_2^3$ | \( 1 - 10 T^{2} - 189 T^{4} - 10 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 3 T - 31 T^{2} + 18 T^{3} + 864 T^{4} + 18 p T^{5} - 31 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - 47 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 9 T + 71 T^{2} - 396 T^{3} + 1812 T^{4} - 396 p T^{5} + 71 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 6 T - 10 T^{2} - 132 T^{3} - 441 T^{4} - 132 p T^{5} - 10 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 9 T + 94 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 49 T^{2} + 660 T^{4} - 49 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 18 T + 218 T^{2} - 1980 T^{3} + 14967 T^{4} - 1980 p T^{5} + 218 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 3 T - 91 T^{2} - 18 T^{3} + 6714 T^{4} - 18 p T^{5} - 91 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 9 T + 17 T^{2} - 486 T^{3} - 4164 T^{4} - 486 p T^{5} + 17 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 27 T + 401 T^{2} + 4266 T^{3} + 36066 T^{4} + 4266 p T^{5} + 401 p^{2} T^{6} + 27 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 9 T + 143 T^{2} - 1044 T^{3} + 10776 T^{4} - 1044 p T^{5} + 143 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 208 T^{2} + 19710 T^{4} - 208 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $C_2^2$ | \( ( 1 - 2 T - 69 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 + T - 83 T^{2} - 74 T^{3} + 736 T^{4} - 74 p T^{5} - 83 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 190 T^{2} + 18051 T^{4} - 190 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 3 T - 163 T^{2} + 18 T^{3} + 20862 T^{4} + 18 p T^{5} - 163 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 13 T + 162 T^{2} + 13 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−9.136779732602358153146252332528, −9.031961126467121765048246567950, −8.761332846228992904031992527398, −8.758011600264658810037086445088, −7.86508457166676044497438649880, −7.41956053095984891250775878257, −7.14233005846698791964978395761, −6.97982584962733298409023531441, −6.53848819121091112833528499951, −6.50667406881112696311230589551, −6.22878800057677037886620362661, −6.12444944575195339898110914673, −5.88868914188935807420632921639, −5.67408968188284297306057447374, −5.01276061802680401246450665826, −4.55532473929117568496186694290, −4.33961987630648039761604936268, −4.28538566094244230463954409208, −3.76191248847907247816191456230, −3.30524484863273526556356751601, −2.99766768252635341737852230372, −2.68727074443064898284934803942, −2.16440665677769000526595818414, −2.15190011609832083644186196865, −0.937759310051651592901227003732,
0.937759310051651592901227003732, 2.15190011609832083644186196865, 2.16440665677769000526595818414, 2.68727074443064898284934803942, 2.99766768252635341737852230372, 3.30524484863273526556356751601, 3.76191248847907247816191456230, 4.28538566094244230463954409208, 4.33961987630648039761604936268, 4.55532473929117568496186694290, 5.01276061802680401246450665826, 5.67408968188284297306057447374, 5.88868914188935807420632921639, 6.12444944575195339898110914673, 6.22878800057677037886620362661, 6.50667406881112696311230589551, 6.53848819121091112833528499951, 6.97982584962733298409023531441, 7.14233005846698791964978395761, 7.41956053095984891250775878257, 7.86508457166676044497438649880, 8.758011600264658810037086445088, 8.761332846228992904031992527398, 9.031961126467121765048246567950, 9.136779732602358153146252332528
