| L(s) = 1 | − 3-s + 5-s + 9-s − 15-s + 2·17-s − 4·23-s + 25-s − 27-s + 6·29-s + 8·31-s − 10·37-s + 6·41-s − 4·43-s + 45-s − 8·47-s − 7·49-s − 2·51-s + 2·53-s + 6·61-s + 12·67-s + 4·69-s + 8·71-s − 6·73-s − 75-s − 8·79-s + 81-s − 12·83-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1/3·9-s − 0.258·15-s + 0.485·17-s − 0.834·23-s + 1/5·25-s − 0.192·27-s + 1.11·29-s + 1.43·31-s − 1.64·37-s + 0.937·41-s − 0.609·43-s + 0.149·45-s − 1.16·47-s − 49-s − 0.280·51-s + 0.274·53-s + 0.768·61-s + 1.46·67-s + 0.481·69-s + 0.949·71-s − 0.702·73-s − 0.115·75-s − 0.900·79-s + 1/9·81-s − 1.31·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.901453835\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.901453835\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 5 | \( 1 - T \) | |
| 13 | \( 1 \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−14.63540458694805, −14.16073887337655, −13.88130579968643, −13.04273915601511, −12.79402028621662, −12.00710151797706, −11.76258308062379, −11.18640939186334, −10.27542858538275, −10.25497022319488, −9.660114905936253, −8.963763386473411, −8.231439212281076, −7.977428001388164, −7.030522503068956, −6.588845224667505, −6.127056471713186, −5.430539188007725, −4.951730422612896, −4.348781241512449, −3.569360942546312, −2.897906296457915, −2.087564178708744, −1.366638664573329, −0.5431315471412454,
0.5431315471412454, 1.366638664573329, 2.087564178708744, 2.897906296457915, 3.569360942546312, 4.348781241512449, 4.951730422612896, 5.430539188007725, 6.127056471713186, 6.588845224667505, 7.030522503068956, 7.977428001388164, 8.231439212281076, 8.963763386473411, 9.660114905936253, 10.25497022319488, 10.27542858538275, 11.18640939186334, 11.76258308062379, 12.00710151797706, 12.79402028621662, 13.04273915601511, 13.88130579968643, 14.16073887337655, 14.63540458694805
