| L(s) = 1 | − 4-s + 7-s − 2·9-s − 2·13-s − 3·16-s − 6·17-s − 8·19-s + 2·25-s − 28-s + 2·36-s − 8·37-s − 6·41-s + 49-s + 2·52-s + 12·53-s − 2·61-s − 2·63-s + 7·64-s + 16·67-s + 6·68-s + 10·73-s + 8·76-s − 5·81-s − 12·83-s − 2·91-s − 2·100-s − 18·101-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 0.377·7-s − 2/3·9-s − 0.554·13-s − 3/4·16-s − 1.45·17-s − 1.83·19-s + 2/5·25-s − 0.188·28-s + 1/3·36-s − 1.31·37-s − 0.937·41-s + 1/7·49-s + 0.277·52-s + 1.64·53-s − 0.256·61-s − 0.251·63-s + 7/8·64-s + 1.95·67-s + 0.727·68-s + 1.17·73-s + 0.917·76-s − 5/9·81-s − 1.31·83-s − 0.209·91-s − 1/5·100-s − 1.79·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 41503 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 41503 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 11 | $C_2$ | \( 1 + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 82 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
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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
−10.00339241750351264117829418170, −9.361465963281960779078601642683, −8.824094859929164621480104989929, −8.473668772774342295169233493067, −8.245014544319781858814417529993, −7.15393507603149142306785349906, −6.82524420906347299159468239871, −6.31344285104076121560165509384, −5.43498540081533097817684177296, −4.93991896528081558905897577985, −4.33805413117707046904059736988, −3.78587422886686534112404405923, −2.58581886128655065343950914699, −2.03742948765031454885572790517, 0,
2.03742948765031454885572790517, 2.58581886128655065343950914699, 3.78587422886686534112404405923, 4.33805413117707046904059736988, 4.93991896528081558905897577985, 5.43498540081533097817684177296, 6.31344285104076121560165509384, 6.82524420906347299159468239871, 7.15393507603149142306785349906, 8.245014544319781858814417529993, 8.473668772774342295169233493067, 8.824094859929164621480104989929, 9.361465963281960779078601642683, 10.00339241750351264117829418170
