| L(s) = 1 | − 5-s − 4·7-s + 6·17-s − 4·19-s + 6·23-s + 25-s + 6·29-s − 10·31-s + 4·35-s + 10·37-s − 6·41-s + 4·43-s − 12·47-s + 9·49-s − 12·53-s + 12·59-s − 10·61-s + 14·67-s + 16·73-s − 8·79-s − 12·83-s − 6·85-s + 6·89-s + 4·95-s − 8·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.51·7-s + 1.45·17-s − 0.917·19-s + 1.25·23-s + 1/5·25-s + 1.11·29-s − 1.79·31-s + 0.676·35-s + 1.64·37-s − 0.937·41-s + 0.609·43-s − 1.75·47-s + 9/7·49-s − 1.64·53-s + 1.56·59-s − 1.28·61-s + 1.71·67-s + 1.87·73-s − 0.900·79-s − 1.31·83-s − 0.650·85-s + 0.635·89-s + 0.410·95-s − 0.812·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 121680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.332063499\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.332063499\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | Isogeny Class over $\mathbf{F}_p$ |
|---|
| bad | 2 | \( 1 \) | |
| 3 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| good | 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 14 T + p T^{2} \) | 1.67.ao |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 16 T + p T^{2} \) | 1.73.aq |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.33583948539757, −12.94253862392979, −12.63346180247773, −12.34335086158927, −11.59367554539286, −11.09225345355810, −10.71633987436577, −10.01542599990411, −9.618863173208560, −9.366259255384829, −8.572955355296569, −8.178533064043448, −7.613239762416501, −6.950237316553925, −6.662817461296170, −6.106837276914478, −5.526272723830802, −4.960575145604299, −4.299930047122071, −3.639697238143752, −3.194859281147814, −2.845026893847636, −1.949762167711729, −1.070202129141691, −0.3979767400199660,
0.3979767400199660, 1.070202129141691, 1.949762167711729, 2.845026893847636, 3.194859281147814, 3.639697238143752, 4.299930047122071, 4.960575145604299, 5.526272723830802, 6.106837276914478, 6.662817461296170, 6.950237316553925, 7.613239762416501, 8.178533064043448, 8.572955355296569, 9.366259255384829, 9.618863173208560, 10.01542599990411, 10.71633987436577, 11.09225345355810, 11.59367554539286, 12.34335086158927, 12.63346180247773, 12.94253862392979, 13.33583948539757
