| L(s) = 1 | − 3-s + 5-s − 7-s + 9-s − 6·13-s − 15-s + 3·17-s + 4·19-s + 21-s − 6·23-s − 4·25-s − 27-s + 6·31-s − 35-s − 6·37-s + 6·39-s + 10·41-s − 11·43-s + 45-s + 9·47-s + 49-s − 3·51-s − 12·53-s − 4·57-s − 7·59-s + 2·61-s − 63-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 0.377·7-s + 1/3·9-s − 1.66·13-s − 0.258·15-s + 0.727·17-s + 0.917·19-s + 0.218·21-s − 1.25·23-s − 4/5·25-s − 0.192·27-s + 1.07·31-s − 0.169·35-s − 0.986·37-s + 0.960·39-s + 1.56·41-s − 1.67·43-s + 0.149·45-s + 1.31·47-s + 1/7·49-s − 0.420·51-s − 1.64·53-s − 0.529·57-s − 0.911·59-s + 0.256·61-s − 0.125·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40656 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40656 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.076033114\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.076033114\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 7 | \( 1 + T \) | |
| 11 | \( 1 \) | |
| good | 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 + 11 T + p T^{2} \) | 1.43.l |
| 47 | \( 1 - 9 T + p T^{2} \) | 1.47.aj |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 + 7 T + p T^{2} \) | 1.59.h |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 9 T + p T^{2} \) | 1.67.aj |
| 71 | \( 1 - 2 T + p T^{2} \) | 1.71.ac |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + 3 T + p T^{2} \) | 1.83.d |
| 89 | \( 1 - 15 T + p T^{2} \) | 1.89.ap |
| 97 | \( 1 + 4 T + p T^{2} \) | 1.97.e |
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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
−14.60223316854859, −14.22668209015570, −13.81394571150132, −13.19505072031631, −12.51657798987784, −12.11415381546625, −11.84421467989534, −11.18877160030653, −10.23197884355457, −10.16921353826587, −9.586120650586588, −9.217248039795524, −8.234456511247677, −7.619753365577593, −7.377178190651642, −6.446175550565381, −6.158948807231636, −5.345186598630952, −5.076418182621852, −4.296351151000952, −3.591091909775096, −2.818036861816253, −2.180260930212278, −1.390686415320193, −0.3931215081634168,
0.3931215081634168, 1.390686415320193, 2.180260930212278, 2.818036861816253, 3.591091909775096, 4.296351151000952, 5.076418182621852, 5.345186598630952, 6.158948807231636, 6.446175550565381, 7.377178190651642, 7.619753365577593, 8.234456511247677, 9.217248039795524, 9.586120650586588, 10.16921353826587, 10.23197884355457, 11.18877160030653, 11.84421467989534, 12.11415381546625, 12.51657798987784, 13.19505072031631, 13.81394571150132, 14.22668209015570, 14.60223316854859
