L(s) = 1 | − 3·4-s + 4·7-s − 2·13-s + 5·16-s + 12·19-s + 25-s − 12·28-s − 20·31-s − 4·37-s − 16·43-s − 2·49-s + 6·52-s + 4·61-s − 3·64-s − 20·67-s − 12·73-s − 36·76-s + 24·79-s − 8·91-s − 28·97-s − 3·100-s − 8·103-s − 4·109-s + 20·112-s − 6·121-s + 60·124-s + 127-s + ⋯ |
L(s) = 1 | − 3/2·4-s + 1.51·7-s − 0.554·13-s + 5/4·16-s + 2.75·19-s + 1/5·25-s − 2.26·28-s − 3.59·31-s − 0.657·37-s − 2.43·43-s − 2/7·49-s + 0.832·52-s + 0.512·61-s − 3/8·64-s − 2.44·67-s − 1.40·73-s − 4.12·76-s + 2.70·79-s − 0.838·91-s − 2.84·97-s − 0.299·100-s − 0.788·103-s − 0.383·109-s + 1.88·112-s − 0.545·121-s + 5.38·124-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 342225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | | \( 1 \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 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
−8.455924495253961688482406234394, −8.069603374513132812350986247673, −7.77509555057385964870974093324, −7.09926588277152631380243196233, −7.01279831140022405844648391421, −5.80203985872581797785224323059, −5.33227627105416183888148278982, −5.17362861953367702283649965368, −4.80291034122253544985154871503, −4.13601102912022186614527931542, −3.48108877573719190338767515003, −3.11999237486850599731326861173, −1.80936597723698845777463745934, −1.37276923078037774007010116691, 0,
1.37276923078037774007010116691, 1.80936597723698845777463745934, 3.11999237486850599731326861173, 3.48108877573719190338767515003, 4.13601102912022186614527931542, 4.80291034122253544985154871503, 5.17362861953367702283649965368, 5.33227627105416183888148278982, 5.80203985872581797785224323059, 7.01279831140022405844648391421, 7.09926588277152631380243196233, 7.77509555057385964870974093324, 8.069603374513132812350986247673, 8.455924495253961688482406234394