L(s) = 1 | − 3-s − 4·5-s + 9-s + 4·15-s + 4·17-s + 2·25-s − 27-s + 12·37-s − 12·41-s + 8·43-s − 4·45-s − 7·49-s − 4·51-s + 8·59-s − 8·67-s − 2·75-s − 16·79-s + 81-s − 8·83-s − 16·85-s − 12·89-s − 36·101-s − 4·109-s − 12·111-s − 6·121-s + 12·123-s + 28·125-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 1.78·5-s + 1/3·9-s + 1.03·15-s + 0.970·17-s + 2/5·25-s − 0.192·27-s + 1.97·37-s − 1.87·41-s + 1.21·43-s − 0.596·45-s − 49-s − 0.560·51-s + 1.04·59-s − 0.977·67-s − 0.230·75-s − 1.80·79-s + 1/9·81-s − 0.878·83-s − 1.73·85-s − 1.27·89-s − 3.58·101-s − 0.383·109-s − 1.13·111-s − 0.545·121-s + 1.08·123-s + 2.50·125-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84672 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84672 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 + T \) |
| 7 | $C_2$ | \( 1 + p T^{2} \) |
good | 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−9.570243823567071698489558383687, −8.882269440513519494115608595002, −8.098990694093691505068092710868, −8.070493008159816865665193495447, −7.48262505518830506099686358492, −7.01507823160614642101417797126, −6.42897107072744719896577758454, −5.68882761687599646139467960885, −5.31393935703508661243014911411, −4.26006602995316384735017711370, −4.25303028692796488061253338187, −3.44517807903584076206897070581, −2.73151126809740041559980819161, −1.33234813740225181579568980351, 0,
1.33234813740225181579568980351, 2.73151126809740041559980819161, 3.44517807903584076206897070581, 4.25303028692796488061253338187, 4.26006602995316384735017711370, 5.31393935703508661243014911411, 5.68882761687599646139467960885, 6.42897107072744719896577758454, 7.01507823160614642101417797126, 7.48262505518830506099686358492, 8.070493008159816865665193495447, 8.098990694093691505068092710868, 8.882269440513519494115608595002, 9.570243823567071698489558383687