L(s) = 1 | − 2·2-s + 3·4-s + 7-s − 4·8-s + 9-s + 2·11-s − 2·14-s + 5·16-s − 2·18-s − 4·22-s − 8·23-s − 6·25-s + 3·28-s + 4·29-s − 6·32-s + 3·36-s − 4·37-s + 6·44-s + 16·46-s + 49-s + 12·50-s − 28·53-s − 4·56-s − 8·58-s + 63-s + 7·64-s + 8·67-s + ⋯ |
L(s) = 1 | − 1.41·2-s + 3/2·4-s + 0.377·7-s − 1.41·8-s + 1/3·9-s + 0.603·11-s − 0.534·14-s + 5/4·16-s − 0.471·18-s − 0.852·22-s − 1.66·23-s − 6/5·25-s + 0.566·28-s + 0.742·29-s − 1.06·32-s + 1/2·36-s − 0.657·37-s + 0.904·44-s + 2.35·46-s + 1/7·49-s + 1.69·50-s − 3.84·53-s − 0.534·56-s − 1.05·58-s + 0.125·63-s + 7/8·64-s + 0.977·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1494108 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1494108 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_1$ | \( 1 - T \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 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
−7.84439006039971069748622971041, −7.49024027339887201227457052610, −6.91394332771191655575545668718, −6.43679449975526542026435018487, −6.17569848665337782312463950999, −5.79323544635467042022952173625, −4.93054069578340184917711747999, −4.72832473817225769292057457440, −3.72716170584453900058590617219, −3.71446826599373971796272627437, −2.83711647406054313257956134718, −1.96943863395594925899870544120, −1.87596574858564202101556695192, −1.00454239585262764496288710387, 0,
1.00454239585262764496288710387, 1.87596574858564202101556695192, 1.96943863395594925899870544120, 2.83711647406054313257956134718, 3.71446826599373971796272627437, 3.72716170584453900058590617219, 4.72832473817225769292057457440, 4.93054069578340184917711747999, 5.79323544635467042022952173625, 6.17569848665337782312463950999, 6.43679449975526542026435018487, 6.91394332771191655575545668718, 7.49024027339887201227457052610, 7.84439006039971069748622971041