| L(s) = 1 | − 3·3-s + 2·4-s − 4·7-s + 6·9-s − 6·12-s − 5·13-s + 8·19-s + 12·21-s + 10·25-s − 9·27-s − 8·28-s − 14·31-s + 12·36-s + 12·37-s + 15·39-s − 13·43-s + 9·49-s − 10·52-s − 24·57-s + 15·61-s − 24·63-s − 8·64-s − 21·67-s + 34·73-s − 30·75-s + 16·76-s + 26·79-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 4-s − 1.51·7-s + 2·9-s − 1.73·12-s − 1.38·13-s + 1.83·19-s + 2.61·21-s + 2·25-s − 1.73·27-s − 1.51·28-s − 2.51·31-s + 2·36-s + 1.97·37-s + 2.40·39-s − 1.98·43-s + 9/7·49-s − 1.38·52-s − 3.17·57-s + 1.92·61-s − 3.02·63-s − 64-s − 2.56·67-s + 3.97·73-s − 3.46·75-s + 1.83·76-s + 2.92·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74529 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74529 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7230065630\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7230065630\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 7 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| 13 | $C_2$ | \( 1 + 5 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 5 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 5 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 19 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.18268868208400242859850209731, −11.52425857137614495837828872205, −11.33803261408755340200972346292, −10.80539758213148015010683110750, −10.40089889788438293275257400370, −9.870708903496518587277158065707, −9.328453358891709060506405346161, −9.300189100969128268134180732686, −8.034897633269054325610519937778, −7.33232074103355178450606162694, −7.10419754443373669064748729414, −6.65428466978253725762091186149, −6.32244065960848665532550068497, −5.55437350422619564645922677295, −5.19047948539486455925557590928, −4.67737010049739350255090035996, −3.57592246721609012130707019413, −3.02829969689145363963164564532, −2.09902232725384002936870075095, −0.71402027238115935797487943558,
0.71402027238115935797487943558, 2.09902232725384002936870075095, 3.02829969689145363963164564532, 3.57592246721609012130707019413, 4.67737010049739350255090035996, 5.19047948539486455925557590928, 5.55437350422619564645922677295, 6.32244065960848665532550068497, 6.65428466978253725762091186149, 7.10419754443373669064748729414, 7.33232074103355178450606162694, 8.034897633269054325610519937778, 9.300189100969128268134180732686, 9.328453358891709060506405346161, 9.870708903496518587277158065707, 10.40089889788438293275257400370, 10.80539758213148015010683110750, 11.33803261408755340200972346292, 11.52425857137614495837828872205, 12.18268868208400242859850209731
