L(s) = 1 | − 4·2-s − 2·3-s + 8·4-s + 8·6-s − 8·8-s − 3·9-s − 16·12-s − 4·16-s + 12·18-s − 2·23-s + 16·24-s − 9·25-s + 14·27-s + 32·32-s − 24·36-s + 6·37-s − 16·41-s + 8·46-s + 8·48-s − 10·49-s + 36·50-s − 56·54-s + 24·61-s − 64·64-s − 14·67-s + 4·69-s − 6·71-s + ⋯ |
L(s) = 1 | − 2.82·2-s − 1.15·3-s + 4·4-s + 3.26·6-s − 2.82·8-s − 9-s − 4.61·12-s − 16-s + 2.82·18-s − 0.417·23-s + 3.26·24-s − 9/5·25-s + 2.69·27-s + 5.65·32-s − 4·36-s + 0.986·37-s − 2.49·41-s + 1.17·46-s + 1.15·48-s − 1.42·49-s + 5.09·50-s − 7.62·54-s + 3.07·61-s − 8·64-s − 1.71·67-s + 0.481·69-s − 0.712·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 644809 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 644809 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 73 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
good | 2 | $C_2$ | \( ( 1 + p T + p T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
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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
−8.366168493001973007262822883179, −7.87113460386926838321975189030, −7.50681548239628598581955812221, −7.00151775708158701065365356556, −6.36261389471308870138602900888, −6.25873307557702221993418985342, −5.55721740976493333762057945802, −5.05582568807268347080999257660, −4.54107515896721559254404612115, −3.73565077084379931334072805578, −2.88381500033411298431745812444, −2.14324784491064065770554543550, −1.56807391045021260794401712953, −0.59933466368807087988368225028, 0,
0.59933466368807087988368225028, 1.56807391045021260794401712953, 2.14324784491064065770554543550, 2.88381500033411298431745812444, 3.73565077084379931334072805578, 4.54107515896721559254404612115, 5.05582568807268347080999257660, 5.55721740976493333762057945802, 6.25873307557702221993418985342, 6.36261389471308870138602900888, 7.00151775708158701065365356556, 7.50681548239628598581955812221, 7.87113460386926838321975189030, 8.366168493001973007262822883179