| L(s) = 1 | + 4-s − 7-s − 2·13-s + 16-s − 4·19-s − 6·25-s − 28-s − 16·31-s + 6·43-s − 6·49-s − 2·52-s + 64-s − 3·73-s − 4·76-s − 4·79-s + 2·91-s − 16·97-s − 6·100-s + 4·103-s + 20·109-s − 112-s − 14·121-s − 16·124-s + 127-s + 131-s + 4·133-s + 137-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 0.377·7-s − 0.554·13-s + 1/4·16-s − 0.917·19-s − 6/5·25-s − 0.188·28-s − 2.87·31-s + 0.914·43-s − 6/7·49-s − 0.277·52-s + 1/8·64-s − 0.351·73-s − 0.458·76-s − 0.450·79-s + 0.209·91-s − 1.62·97-s − 3/5·100-s + 0.394·103-s + 1.91·109-s − 0.0944·112-s − 1.27·121-s − 1.43·124-s + 0.0887·127-s + 0.0873·131-s + 0.346·133-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 165564 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165564 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | | \( 1 \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + p T^{2} ) \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 2 T + p T^{2} ) \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−9.008396983836892003462418239910, −8.579270692429305602137962801016, −7.88076985901541131314806919086, −7.44790995274526122919965355597, −7.18004920453287839454157425275, −6.45670620902883939975408832444, −6.08476625425280506349604022454, −5.50729342835208729568171099973, −5.04166083260593555185233584254, −4.14637548913859065119247396577, −3.79378613255892486142834173555, −3.02925396898687861194049938184, −2.25895213588299889227053061618, −1.67449141677278248735177683986, 0,
1.67449141677278248735177683986, 2.25895213588299889227053061618, 3.02925396898687861194049938184, 3.79378613255892486142834173555, 4.14637548913859065119247396577, 5.04166083260593555185233584254, 5.50729342835208729568171099973, 6.08476625425280506349604022454, 6.45670620902883939975408832444, 7.18004920453287839454157425275, 7.44790995274526122919965355597, 7.88076985901541131314806919086, 8.579270692429305602137962801016, 9.008396983836892003462418239910
