| L(s) = 1 | + 2-s + 5-s + 7-s − 8-s + 10-s − 11-s + 14-s − 16-s + 4·17-s + 2·19-s − 22-s − 23-s + 25-s − 31-s + 4·34-s + 35-s − 2·37-s + 2·38-s − 40-s + 41-s − 46-s + 50-s − 55-s − 56-s − 62-s + 64-s + 70-s + ⋯ |
| L(s) = 1 | + 2-s + 5-s + 7-s − 8-s + 10-s − 11-s + 14-s − 16-s + 4·17-s + 2·19-s − 22-s − 23-s + 25-s − 31-s + 4·34-s + 35-s − 2·37-s + 2·38-s − 40-s + 41-s − 46-s + 50-s − 55-s − 56-s − 62-s + 64-s + 70-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5143824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5143824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.794959444\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.794959444\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( 1 - T + T^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 5 | $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 | $C_1$ | \( ( 1 - T )^{4} \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 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_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 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
−9.633467101347565511154702836330, −9.206498672716002321955975821388, −8.386964245086595000936451757591, −8.196326751296965268077681621857, −7.925445301327123315978242036883, −7.49958351367229377470887375163, −7.09217965295457934381414450408, −6.67632847316378803248875895481, −5.74277004547041211131294746063, −5.68826070690439759320572913694, −5.51453117300928735154655764194, −5.07041090306899905344249599317, −5.02979612245365128530448139990, −4.06108478928931631392451988095, −3.69450541689916265986934381579, −3.20879248805190118779231409645, −2.92511609828972627507093950251, −2.32213462980658126936753311184, −1.39950496191664247000278156541, −1.23122936992700661333303101077,
1.23122936992700661333303101077, 1.39950496191664247000278156541, 2.32213462980658126936753311184, 2.92511609828972627507093950251, 3.20879248805190118779231409645, 3.69450541689916265986934381579, 4.06108478928931631392451988095, 5.02979612245365128530448139990, 5.07041090306899905344249599317, 5.51453117300928735154655764194, 5.68826070690439759320572913694, 5.74277004547041211131294746063, 6.67632847316378803248875895481, 7.09217965295457934381414450408, 7.49958351367229377470887375163, 7.925445301327123315978242036883, 8.196326751296965268077681621857, 8.386964245086595000936451757591, 9.206498672716002321955975821388, 9.633467101347565511154702836330
