L(s) = 1 | − 2·3-s + 4-s − 4·7-s + 9-s − 2·12-s + 16-s − 2·17-s + 8·21-s − 10·25-s + 4·27-s − 4·28-s + 36-s − 8·37-s + 12·41-s + 16·43-s − 2·48-s + 9·49-s + 4·51-s − 4·63-s + 64-s + 16·67-s − 2·68-s + 20·75-s + 16·79-s − 11·81-s + 8·84-s − 12·89-s + ⋯ |
L(s) = 1 | − 1.15·3-s + 1/2·4-s − 1.51·7-s + 1/3·9-s − 0.577·12-s + 1/4·16-s − 0.485·17-s + 1.74·21-s − 2·25-s + 0.769·27-s − 0.755·28-s + 1/6·36-s − 1.31·37-s + 1.87·41-s + 2.43·43-s − 0.288·48-s + 9/7·49-s + 0.560·51-s − 0.503·63-s + 1/8·64-s + 1.95·67-s − 0.242·68-s + 2.30·75-s + 1.80·79-s − 1.22·81-s + 0.872·84-s − 1.27·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 509796 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 509796 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 7 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| 17 | $C_1$ | \( ( 1 + T )^{2} \) |
good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−8.160618185897863834823885381010, −7.66308857193970764960259229760, −7.31851367584369797076036436129, −6.65330731261113455462227374582, −6.45610714582403038230359552601, −5.99911855171913780844071238816, −5.62338766094370476972651016075, −5.24003384213634180871929880044, −4.39135887137036440164240725215, −3.90229547122613769308947697962, −3.44317413810209705275397876483, −2.58010997151442346007022994115, −2.20036496479885516041956568052, −0.942238784664548367564857344717, 0,
0.942238784664548367564857344717, 2.20036496479885516041956568052, 2.58010997151442346007022994115, 3.44317413810209705275397876483, 3.90229547122613769308947697962, 4.39135887137036440164240725215, 5.24003384213634180871929880044, 5.62338766094370476972651016075, 5.99911855171913780844071238816, 6.45610714582403038230359552601, 6.65330731261113455462227374582, 7.31851367584369797076036436129, 7.66308857193970764960259229760, 8.160618185897863834823885381010