| L(s) = 1 | − 4·3-s + 6·9-s + 10·11-s + 10·17-s + 2·19-s + 25-s + 4·27-s − 40·33-s + 12·41-s − 2·43-s − 3·49-s − 40·51-s − 8·57-s − 12·59-s − 16·67-s + 18·73-s − 4·75-s − 37·81-s + 8·83-s + 8·89-s − 24·97-s + 60·99-s − 12·107-s + 36·113-s + 53·121-s − 48·123-s + 127-s + ⋯ |
| L(s) = 1 | − 2.30·3-s + 2·9-s + 3.01·11-s + 2.42·17-s + 0.458·19-s + 1/5·25-s + 0.769·27-s − 6.96·33-s + 1.87·41-s − 0.304·43-s − 3/7·49-s − 5.60·51-s − 1.05·57-s − 1.56·59-s − 1.95·67-s + 2.10·73-s − 0.461·75-s − 4.11·81-s + 0.878·83-s + 0.847·89-s − 2.43·97-s + 6.03·99-s − 1.16·107-s + 3.38·113-s + 4.81·121-s − 4.32·123-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23658496 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23658496 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.970270104\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.970270104\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 3 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 + 23 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 98 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
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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
−8.408955934204590632298213978341, −8.067205164676451621213300957554, −7.56091010996316573565695497531, −7.17772070487467020282965437766, −6.90242670636169450566330297932, −6.48121400310534845326283856079, −6.04532390316239214897641829065, −6.03555115459770356723542639384, −5.73048567047010105943877079604, −5.25762982876777096231523374680, −4.77916638273426597161291059640, −4.60611409192651308942943323164, −3.95544101904848369680286539119, −3.73731650126429940795228045054, −3.12131019385518060962220064642, −2.87847179444535643173325927741, −1.69915756574590723725870940623, −1.45119405589658494255691524224, −0.76117520015341031534849437195, −0.73517404529325591849329543340,
0.73517404529325591849329543340, 0.76117520015341031534849437195, 1.45119405589658494255691524224, 1.69915756574590723725870940623, 2.87847179444535643173325927741, 3.12131019385518060962220064642, 3.73731650126429940795228045054, 3.95544101904848369680286539119, 4.60611409192651308942943323164, 4.77916638273426597161291059640, 5.25762982876777096231523374680, 5.73048567047010105943877079604, 6.03555115459770356723542639384, 6.04532390316239214897641829065, 6.48121400310534845326283856079, 6.90242670636169450566330297932, 7.17772070487467020282965437766, 7.56091010996316573565695497531, 8.067205164676451621213300957554, 8.408955934204590632298213978341
