L(s) = 1 | − 4·3-s + 4-s − 8·7-s + 6·9-s − 4·12-s + 16-s − 17-s − 8·19-s + 32·21-s − 5·25-s + 4·27-s − 8·28-s + 6·36-s − 8·37-s − 4·48-s + 34·49-s + 4·51-s + 32·57-s − 48·63-s + 64-s − 68-s + 4·73-s + 20·75-s − 8·76-s − 37·81-s + 32·84-s − 12·89-s + ⋯ |
L(s) = 1 | − 2.30·3-s + 1/2·4-s − 3.02·7-s + 2·9-s − 1.15·12-s + 1/4·16-s − 0.242·17-s − 1.83·19-s + 6.98·21-s − 25-s + 0.769·27-s − 1.51·28-s + 36-s − 1.31·37-s − 0.577·48-s + 34/7·49-s + 0.560·51-s + 4.23·57-s − 6.04·63-s + 1/8·64-s − 0.121·68-s + 0.468·73-s + 2.30·75-s − 0.917·76-s − 4.11·81-s + 3.49·84-s − 1.27·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 491300 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 491300 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | $C_2$ | \( 1 + p T^{2} \) |
| 17 | $C_1$ | \( 1 + T \) |
good | 3 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
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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
−8.434721631247958880519323524946, −7.54625870597452603613853398245, −6.97765460885046698808229959612, −6.65330731261113455462227374582, −6.39126314971633693713494192915, −5.99911855171913780844071238816, −5.88165569807091607176477515994, −5.21315012084683261869529031894, −4.61516560042655420117257789936, −3.90229547122613769308947697962, −3.39756711777290923826252356515, −2.81090884111852531330067525726, −2.03988638187729415468870477153, −0.56239664780041394316008252541, 0,
0.56239664780041394316008252541, 2.03988638187729415468870477153, 2.81090884111852531330067525726, 3.39756711777290923826252356515, 3.90229547122613769308947697962, 4.61516560042655420117257789936, 5.21315012084683261869529031894, 5.88165569807091607176477515994, 5.99911855171913780844071238816, 6.39126314971633693713494192915, 6.65330731261113455462227374582, 6.97765460885046698808229959612, 7.54625870597452603613853398245, 8.434721631247958880519323524946