| L(s) = 1 | − 5-s + 7-s − 11-s − 13-s + 5·17-s + 3·23-s + 25-s + 2·29-s + 6·31-s − 35-s + 37-s + 7·41-s − 2·43-s − 6·47-s − 6·49-s + 5·53-s + 55-s − 4·59-s − 7·61-s + 65-s − 12·67-s − 71-s + 10·73-s − 77-s − 11·79-s − 12·83-s − 5·85-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.377·7-s − 0.301·11-s − 0.277·13-s + 1.21·17-s + 0.625·23-s + 1/5·25-s + 0.371·29-s + 1.07·31-s − 0.169·35-s + 0.164·37-s + 1.09·41-s − 0.304·43-s − 0.875·47-s − 6/7·49-s + 0.686·53-s + 0.134·55-s − 0.520·59-s − 0.896·61-s + 0.124·65-s − 1.46·67-s − 0.118·71-s + 1.17·73-s − 0.113·77-s − 1.23·79-s − 1.31·83-s − 0.542·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.943719887\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.943719887\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 13 | \( 1 + T \) | |
| good | 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 + T + p T^{2} \) | 1.11.b |
| 17 | \( 1 - 5 T + p T^{2} \) | 1.17.af |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 - T + p T^{2} \) | 1.37.ab |
| 41 | \( 1 - 7 T + p T^{2} \) | 1.41.ah |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 - 5 T + p T^{2} \) | 1.53.af |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 7 T + p T^{2} \) | 1.61.h |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + T + p T^{2} \) | 1.71.b |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 + 11 T + p T^{2} \) | 1.79.l |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 13 T + p T^{2} \) | 1.89.an |
| 97 | \( 1 + T + p T^{2} \) | 1.97.b |
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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
−7.76023196326668453875270868040, −7.16851705630372958395568706026, −6.33379709200546649359935910353, −5.62934638711337750122578666526, −4.84445386484758626916511511540, −4.36104693383824063435695469895, −3.30266601623606956186214863311, −2.79825271995589007003021293339, −1.63814347259597507594005992257, −0.69138921082913773526262386929,
0.69138921082913773526262386929, 1.63814347259597507594005992257, 2.79825271995589007003021293339, 3.30266601623606956186214863311, 4.36104693383824063435695469895, 4.84445386484758626916511511540, 5.62934638711337750122578666526, 6.33379709200546649359935910353, 7.16851705630372958395568706026, 7.76023196326668453875270868040
