| L(s) = 1 | − 5-s − 4·7-s + 4·11-s − 2·13-s + 6·17-s − 6·19-s + 6·23-s + 25-s + 2·29-s + 4·31-s + 4·35-s − 8·37-s + 8·41-s − 43-s + 6·47-s + 9·49-s − 6·53-s − 4·55-s + 10·61-s + 2·65-s + 12·67-s − 16·71-s + 16·73-s − 16·77-s + 4·79-s + 6·83-s − 6·85-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.51·7-s + 1.20·11-s − 0.554·13-s + 1.45·17-s − 1.37·19-s + 1.25·23-s + 1/5·25-s + 0.371·29-s + 0.718·31-s + 0.676·35-s − 1.31·37-s + 1.24·41-s − 0.152·43-s + 0.875·47-s + 9/7·49-s − 0.824·53-s − 0.539·55-s + 1.28·61-s + 0.248·65-s + 1.46·67-s − 1.89·71-s + 1.87·73-s − 1.82·77-s + 0.450·79-s + 0.658·83-s − 0.650·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 123840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 123840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.098450327\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.098450327\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 43 | \( 1 + T \) | |
| good | 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 - 8 T + p T^{2} \) | 1.41.ai |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + 16 T + p T^{2} \) | 1.71.q |
| 73 | \( 1 - 16 T + p T^{2} \) | 1.73.aq |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 97 | \( 1 + 18 T + p T^{2} \) | 1.97.s |
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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
−13.53063606034897, −12.82198547360593, −12.57617638935274, −12.19318528015222, −11.79509731162131, −11.02770963507632, −10.66459821197353, −10.02591873135861, −9.621576154073589, −9.263298461470572, −8.623907254587132, −8.250565609508538, −7.477047864398301, −6.866794509697475, −6.771953719293970, −6.059893390602652, −5.631059615786737, −4.819046124152477, −4.319294817005047, −3.600447742112419, −3.357082283436828, −2.700743534808692, −1.988406534910153, −0.9810956575279883, −0.5384057508709303,
0.5384057508709303, 0.9810956575279883, 1.988406534910153, 2.700743534808692, 3.357082283436828, 3.600447742112419, 4.319294817005047, 4.819046124152477, 5.631059615786737, 6.059893390602652, 6.771953719293970, 6.866794509697475, 7.477047864398301, 8.250565609508538, 8.623907254587132, 9.263298461470572, 9.621576154073589, 10.02591873135861, 10.66459821197353, 11.02770963507632, 11.79509731162131, 12.19318528015222, 12.57617638935274, 12.82198547360593, 13.53063606034897
