| L(s) = 1 | − 3-s − 4·7-s − 2·9-s + 4·13-s + 6·17-s − 8·19-s + 4·21-s + 5·27-s − 6·29-s + 5·31-s − 11·37-s − 4·39-s − 6·41-s + 43-s + 6·47-s + 9·49-s − 6·51-s + 3·53-s + 8·57-s − 9·59-s − 5·61-s + 8·63-s + 10·67-s + 6·71-s − 7·73-s + 79-s + 81-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.51·7-s − 2/3·9-s + 1.10·13-s + 1.45·17-s − 1.83·19-s + 0.872·21-s + 0.962·27-s − 1.11·29-s + 0.898·31-s − 1.80·37-s − 0.640·39-s − 0.937·41-s + 0.152·43-s + 0.875·47-s + 9/7·49-s − 0.840·51-s + 0.412·53-s + 1.05·57-s − 1.17·59-s − 0.640·61-s + 1.00·63-s + 1.22·67-s + 0.712·71-s − 0.819·73-s + 0.112·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 126400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 79 | \( 1 - T \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 5 T + p T^{2} \) | 1.31.af |
| 37 | \( 1 + 11 T + p T^{2} \) | 1.37.l |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 - 3 T + p T^{2} \) | 1.53.ad |
| 59 | \( 1 + 9 T + p T^{2} \) | 1.59.j |
| 61 | \( 1 + 5 T + p T^{2} \) | 1.61.f |
| 67 | \( 1 - 10 T + p T^{2} \) | 1.67.ak |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 + 7 T + p T^{2} \) | 1.73.h |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 11 T + p T^{2} \) | 1.97.al |
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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
−13.65877195739513, −13.27626183557372, −12.67987786787708, −12.28024881560329, −12.02506490045951, −11.28695313797905, −10.76320673369512, −10.45882701042428, −9.987868314932625, −9.384684173824854, −8.846317798151049, −8.444455073443695, −7.966509721029714, −7.084299992523252, −6.742126101938427, −6.141280894779321, −5.888556831779910, −5.417165949315728, −4.678640084125677, −3.900371740361381, −3.468203902160247, −3.062505040651863, −2.285743244807531, −1.501637091055535, −0.6270473643939869, 0,
0.6270473643939869, 1.501637091055535, 2.285743244807531, 3.062505040651863, 3.468203902160247, 3.900371740361381, 4.678640084125677, 5.417165949315728, 5.888556831779910, 6.141280894779321, 6.742126101938427, 7.084299992523252, 7.966509721029714, 8.444455073443695, 8.846317798151049, 9.384684173824854, 9.987868314932625, 10.45882701042428, 10.76320673369512, 11.28695313797905, 12.02506490045951, 12.28024881560329, 12.67987786787708, 13.27626183557372, 13.65877195739513
