| L(s) = 1 | − 4-s − 2·5-s + 7-s + 11-s − 2·17-s + 2·19-s + 2·20-s + 23-s + 25-s − 28-s − 2·35-s + 43-s − 44-s + 47-s + 49-s − 2·55-s + 61-s + 64-s + 2·68-s − 2·73-s − 2·76-s + 77-s − 2·83-s + 4·85-s − 92-s − 4·95-s − 100-s + ⋯ |
| L(s) = 1 | − 4-s − 2·5-s + 7-s + 11-s − 2·17-s + 2·19-s + 2·20-s + 23-s + 25-s − 28-s − 2·35-s + 43-s − 44-s + 47-s + 49-s − 2·55-s + 61-s + 64-s + 2·68-s − 2·73-s − 2·76-s + 77-s − 2·83-s + 4·85-s − 92-s − 4·95-s − 100-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2368521 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2368521 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6061662413\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6061662413\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 47 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 97 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.991254375442355384779665473360, −9.058530300766132568082353512907, −9.054462350221225449902312093549, −8.710558031456455191830366852290, −8.346198557202789518640600511744, −7.84820460346061225194974113133, −7.46121563503960108225571751684, −7.08203191634842699397075795090, −7.01425878384551667760183793133, −6.14373512733645951645331919850, −5.67640108097000367348816951767, −5.07712747365096459048640149400, −4.72118727351530114599418890414, −4.28821936830943606031486906638, −4.07023875252508139076595206675, −3.69854526041394775218606109420, −3.04431156110291739140011353220, −2.38859209161679823867295914143, −1.47346561983698824087167778859, −0.69206603536665633115030562796,
0.69206603536665633115030562796, 1.47346561983698824087167778859, 2.38859209161679823867295914143, 3.04431156110291739140011353220, 3.69854526041394775218606109420, 4.07023875252508139076595206675, 4.28821936830943606031486906638, 4.72118727351530114599418890414, 5.07712747365096459048640149400, 5.67640108097000367348816951767, 6.14373512733645951645331919850, 7.01425878384551667760183793133, 7.08203191634842699397075795090, 7.46121563503960108225571751684, 7.84820460346061225194974113133, 8.346198557202789518640600511744, 8.710558031456455191830366852290, 9.054462350221225449902312093549, 9.058530300766132568082353512907, 9.991254375442355384779665473360
