L(s) = 1 | + 3-s − 3·4-s − 4·5-s − 7-s + 9-s − 3·12-s − 4·15-s + 5·16-s − 12·17-s + 12·20-s − 21-s + 2·25-s + 27-s + 3·28-s + 4·35-s − 3·36-s + 12·37-s + 4·41-s − 8·43-s − 4·45-s + 5·48-s + 49-s − 12·51-s + 24·59-s + 12·60-s − 63-s − 3·64-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 3/2·4-s − 1.78·5-s − 0.377·7-s + 1/3·9-s − 0.866·12-s − 1.03·15-s + 5/4·16-s − 2.91·17-s + 2.68·20-s − 0.218·21-s + 2/5·25-s + 0.192·27-s + 0.566·28-s + 0.676·35-s − 1/2·36-s + 1.97·37-s + 0.624·41-s − 1.21·43-s − 0.596·45-s + 0.721·48-s + 1/7·49-s − 1.68·51-s + 3.12·59-s + 1.54·60-s − 0.125·63-s − 3/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9261 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9261 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 - T \) |
| 7 | $C_1$ | \( 1 + T \) |
good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 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
−11.55595081754559805664353277943, −10.92437047522813718231601421137, −9.800766905340478341584155035885, −9.724661210726173973340366981146, −8.672365196683144779961098572501, −8.609463409035164758983775505912, −8.123916451859490481483222873179, −7.25047783802838427330028450541, −6.81422856027217448872168795499, −5.73267711633132091527857634664, −4.61914309368534933978095799917, −4.13559084050773741974089362056, −3.94328917656394473634657471263, −2.64839746138688614832708523229, 0,
2.64839746138688614832708523229, 3.94328917656394473634657471263, 4.13559084050773741974089362056, 4.61914309368534933978095799917, 5.73267711633132091527857634664, 6.81422856027217448872168795499, 7.25047783802838427330028450541, 8.123916451859490481483222873179, 8.609463409035164758983775505912, 8.672365196683144779961098572501, 9.724661210726173973340366981146, 9.800766905340478341584155035885, 10.92437047522813718231601421137, 11.55595081754559805664353277943