| L(s) = 1 | + 2-s − 4-s + 7-s − 3·8-s + 14-s − 16-s − 8·19-s − 6·25-s − 28-s + 4·29-s + 5·32-s + 12·37-s − 8·38-s + 49-s − 6·50-s − 12·53-s − 3·56-s + 4·58-s + 24·59-s + 7·64-s + 12·74-s + 8·76-s − 24·83-s + 98-s + 6·100-s − 16·103-s − 12·106-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s + 0.377·7-s − 1.06·8-s + 0.267·14-s − 1/4·16-s − 1.83·19-s − 6/5·25-s − 0.188·28-s + 0.742·29-s + 0.883·32-s + 1.97·37-s − 1.29·38-s + 1/7·49-s − 0.848·50-s − 1.64·53-s − 0.400·56-s + 0.525·58-s + 3.12·59-s + 7/8·64-s + 1.39·74-s + 0.917·76-s − 2.63·83-s + 0.101·98-s + 3/5·100-s − 1.57·103-s − 1.16·106-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 444528 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 444528 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 - T + p T^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_1$ | \( 1 - T \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−8.351153010775658504468507691128, −8.123916451859490481483222873179, −7.45452630662509700459332399892, −6.81422856027217448872168795499, −6.43539728173776997986575871832, −5.76945209288606009941542010236, −5.73267711633132091527857634664, −4.75834670654208540745088589071, −4.61914309368534933978095799917, −3.94328917656394473634657471263, −3.71279869000131289976682575519, −2.64839746138688614832708523229, −2.38203049551357769485459868076, −1.27003349896142462152543818564, 0,
1.27003349896142462152543818564, 2.38203049551357769485459868076, 2.64839746138688614832708523229, 3.71279869000131289976682575519, 3.94328917656394473634657471263, 4.61914309368534933978095799917, 4.75834670654208540745088589071, 5.73267711633132091527857634664, 5.76945209288606009941542010236, 6.43539728173776997986575871832, 6.81422856027217448872168795499, 7.45452630662509700459332399892, 8.123916451859490481483222873179, 8.351153010775658504468507691128
