| L(s) = 1 | + 3-s + 5-s + 4·7-s + 9-s − 6·11-s + 4·13-s + 15-s + 2·19-s + 4·21-s + 25-s + 27-s + 6·29-s + 4·31-s − 6·33-s + 4·35-s − 2·37-s + 4·39-s − 41-s − 4·43-s + 45-s + 9·49-s − 12·53-s − 6·55-s + 2·57-s − 2·61-s + 4·63-s + 4·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1.51·7-s + 1/3·9-s − 1.80·11-s + 1.10·13-s + 0.258·15-s + 0.458·19-s + 0.872·21-s + 1/5·25-s + 0.192·27-s + 1.11·29-s + 0.718·31-s − 1.04·33-s + 0.676·35-s − 0.328·37-s + 0.640·39-s − 0.156·41-s − 0.609·43-s + 0.149·45-s + 9/7·49-s − 1.64·53-s − 0.809·55-s + 0.264·57-s − 0.256·61-s + 0.503·63-s + 0.496·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 39360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.309767347\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.309767347\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 41 | \( 1 + T \) | |
| good | 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 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.ai |
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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
−14.72614613646584, −14.15448108119573, −13.84188767422231, −13.37603811185052, −12.88275307641873, −12.24445241556830, −11.59242357366719, −11.00078030849898, −10.64859100684329, −10.12103516522487, −9.546684898420731, −8.740012083731008, −8.274993259325835, −8.023914893398905, −7.494760928950966, −6.685336393001988, −6.047364091477224, −5.261092780868472, −4.960993977930818, −4.382184337074274, −3.429885172188533, −2.855468835478526, −2.147943217844082, −1.557769333710666, −0.7521925414166527,
0.7521925414166527, 1.557769333710666, 2.147943217844082, 2.855468835478526, 3.429885172188533, 4.382184337074274, 4.960993977930818, 5.261092780868472, 6.047364091477224, 6.685336393001988, 7.494760928950966, 8.023914893398905, 8.274993259325835, 8.740012083731008, 9.546684898420731, 10.12103516522487, 10.64859100684329, 11.00078030849898, 11.59242357366719, 12.24445241556830, 12.88275307641873, 13.37603811185052, 13.84188767422231, 14.15448108119573, 14.72614613646584
