| L(s) = 1 | + 2·2-s − 4-s − 7-s − 8·8-s − 8·11-s − 2·14-s − 7·16-s − 16·22-s − 6·25-s + 28-s + 4·29-s + 14·32-s + 12·37-s − 8·43-s + 8·44-s + 49-s − 12·50-s − 12·53-s + 8·56-s + 8·58-s + 35·64-s + 8·67-s + 24·74-s + 8·77-s − 32·79-s − 16·86-s + 64·88-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1/2·4-s − 0.377·7-s − 2.82·8-s − 2.41·11-s − 0.534·14-s − 7/4·16-s − 3.41·22-s − 6/5·25-s + 0.188·28-s + 0.742·29-s + 2.47·32-s + 1.97·37-s − 1.21·43-s + 1.20·44-s + 1/7·49-s − 1.69·50-s − 1.64·53-s + 1.06·56-s + 1.05·58-s + 35/8·64-s + 0.977·67-s + 2.78·74-s + 0.911·77-s − 3.60·79-s − 1.72·86-s + 6.82·88-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27783 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27783 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 7 | $C_1$ | \( 1 + T \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 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
−10.18331996404873608095625884352, −9.800766905340478341584155035885, −9.457705101580370517739336150901, −8.609463409035164758983775505912, −8.123916451859490481483222873179, −7.81295430367747747098376616892, −6.81422856027217448872168795499, −5.94607665445287604157300008429, −5.73267711633132091527857634664, −5.02709416296405066539370067618, −4.61914309368534933978095799917, −3.94328917656394473634657471263, −3.05422074105458389777226041971, −2.64839746138688614832708523229, 0,
2.64839746138688614832708523229, 3.05422074105458389777226041971, 3.94328917656394473634657471263, 4.61914309368534933978095799917, 5.02709416296405066539370067618, 5.73267711633132091527857634664, 5.94607665445287604157300008429, 6.81422856027217448872168795499, 7.81295430367747747098376616892, 8.123916451859490481483222873179, 8.609463409035164758983775505912, 9.457705101580370517739336150901, 9.800766905340478341584155035885, 10.18331996404873608095625884352
