L(s) = 1 | + 4-s − 8·7-s − 3·9-s − 4·13-s + 16-s − 4·19-s + 6·25-s − 8·28-s − 3·36-s − 8·37-s + 20·43-s + 34·49-s − 4·52-s − 16·61-s + 24·63-s + 64-s − 20·67-s + 12·73-s − 4·76-s − 24·79-s + 9·81-s + 32·91-s + 12·97-s + 6·100-s − 16·103-s − 8·112-s + 12·117-s + ⋯ |
L(s) = 1 | + 1/2·4-s − 3.02·7-s − 9-s − 1.10·13-s + 1/4·16-s − 0.917·19-s + 6/5·25-s − 1.51·28-s − 1/2·36-s − 1.31·37-s + 3.04·43-s + 34/7·49-s − 0.554·52-s − 2.04·61-s + 3.02·63-s + 1/8·64-s − 2.44·67-s + 1.40·73-s − 0.458·76-s − 2.70·79-s + 81-s + 3.35·91-s + 1.21·97-s + 3/5·100-s − 1.57·103-s − 0.755·112-s + 1.10·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19044 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19044 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + p T^{2} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
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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
−10.68153496695016269554639136746, −10.08495250148779758795099695515, −9.594811427439617650906242640583, −8.971935452602447588346290189052, −8.842466278765578735147133804346, −7.68457323927771212325575411964, −7.10508922175356728922407799432, −6.69441913641732873591204403077, −5.98394421392704393336498598879, −5.84432417959727908710252168629, −4.65850975177844887385319927784, −3.67221876996996920985604231839, −2.90101747667895451160022461046, −2.63232737474552247848840602025, 0,
2.63232737474552247848840602025, 2.90101747667895451160022461046, 3.67221876996996920985604231839, 4.65850975177844887385319927784, 5.84432417959727908710252168629, 5.98394421392704393336498598879, 6.69441913641732873591204403077, 7.10508922175356728922407799432, 7.68457323927771212325575411964, 8.842466278765578735147133804346, 8.971935452602447588346290189052, 9.594811427439617650906242640583, 10.08495250148779758795099695515, 10.68153496695016269554639136746