| L(s) = 1 | − 3-s + 5-s − 7-s + 9-s − 3·13-s − 15-s + 7·19-s + 21-s − 6·23-s − 4·25-s − 27-s + 9·29-s − 35-s − 3·37-s + 3·39-s − 8·41-s + 10·43-s + 45-s − 3·47-s + 49-s + 6·53-s − 7·57-s − 7·59-s − 10·61-s − 63-s − 3·65-s + 3·67-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 0.377·7-s + 1/3·9-s − 0.832·13-s − 0.258·15-s + 1.60·19-s + 0.218·21-s − 1.25·23-s − 4/5·25-s − 0.192·27-s + 1.67·29-s − 0.169·35-s − 0.493·37-s + 0.480·39-s − 1.24·41-s + 1.52·43-s + 0.149·45-s − 0.437·47-s + 1/7·49-s + 0.824·53-s − 0.927·57-s − 0.911·59-s − 1.28·61-s − 0.125·63-s − 0.372·65-s + 0.366·67-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
| \(L(1)\) |
\(\approx\) |
\(1.447276014\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.447276014\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 3 T + p T^{2} \) | 1.13.d |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 7 T + p T^{2} \) | 1.19.ah |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 9 T + p T^{2} \) | 1.29.aj |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 3 T + p T^{2} \) | 1.37.d |
| 41 | \( 1 + 8 T + p T^{2} \) | 1.41.i |
| 43 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 7 T + p T^{2} \) | 1.59.h |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 3 T + p T^{2} \) | 1.67.ad |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 7 T + p T^{2} \) | 1.73.ah |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−14.74990725316344, −13.94730311002712, −13.86952388205903, −13.35612649216144, −12.37684636137526, −12.19573967723826, −11.85948595200452, −11.10508802396386, −10.41887643701823, −10.09642799546675, −9.456037397365464, −9.290404874480213, −8.096414924750199, −7.954335524057939, −7.014670032685497, −6.741581082529726, −5.921512173515063, −5.570601152356305, −4.936745659829454, −4.328322031792479, −3.562012068477923, −2.867590374444973, −2.157259922853032, −1.356601321826505, −0.4672261862835762,
0.4672261862835762, 1.356601321826505, 2.157259922853032, 2.867590374444973, 3.562012068477923, 4.328322031792479, 4.936745659829454, 5.570601152356305, 5.921512173515063, 6.741581082529726, 7.014670032685497, 7.954335524057939, 8.096414924750199, 9.290404874480213, 9.456037397365464, 10.09642799546675, 10.41887643701823, 11.10508802396386, 11.85948595200452, 12.19573967723826, 12.37684636137526, 13.35612649216144, 13.86952388205903, 13.94730311002712, 14.74990725316344
