L(s) = 1 | − 2·3-s + 3·9-s + 4·11-s − 4·13-s − 6·25-s − 4·27-s + 12·29-s − 8·33-s + 8·39-s − 14·49-s + 8·59-s − 4·61-s − 8·67-s + 12·75-s − 16·79-s + 5·81-s − 24·87-s − 12·89-s + 4·97-s + 12·99-s − 36·101-s − 4·109-s + 36·113-s − 12·117-s + 5·121-s + 127-s + 131-s + ⋯ |
L(s) = 1 | − 1.15·3-s + 9-s + 1.20·11-s − 1.10·13-s − 6/5·25-s − 0.769·27-s + 2.22·29-s − 1.39·33-s + 1.28·39-s − 2·49-s + 1.04·59-s − 0.512·61-s − 0.977·67-s + 1.38·75-s − 1.80·79-s + 5/9·81-s − 2.57·87-s − 1.27·89-s + 0.406·97-s + 1.20·99-s − 3.58·101-s − 0.383·109-s + 3.38·113-s − 1.10·117-s + 5/11·121-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 557568 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 557568 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−8.098990694093691505068092710868, −7.82048755073253042108636697295, −7.04619174160680993198292873454, −6.86309261643108304364881933941, −6.42897107072744719896577758454, −5.94195259388428776855182860466, −5.49024592405004234969144901480, −4.94454899743565935082637503624, −4.33990220358944048865857917052, −4.25303028692796488061253338187, −3.32045065868348143323558177944, −2.71136660887374010040841871572, −1.80957061896546408108660751255, −1.14203459290393266350032133514, 0,
1.14203459290393266350032133514, 1.80957061896546408108660751255, 2.71136660887374010040841871572, 3.32045065868348143323558177944, 4.25303028692796488061253338187, 4.33990220358944048865857917052, 4.94454899743565935082637503624, 5.49024592405004234969144901480, 5.94195259388428776855182860466, 6.42897107072744719896577758454, 6.86309261643108304364881933941, 7.04619174160680993198292873454, 7.82048755073253042108636697295, 8.098990694093691505068092710868