| L(s) = 1 | − 5·7-s − 5·13-s + 8·19-s − 4·31-s − 11·37-s − 5·43-s + 18·49-s − 61-s − 11·67-s + 10·73-s − 13·79-s + 25·91-s − 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 1.88·7-s − 1.38·13-s + 1.83·19-s − 0.718·31-s − 1.80·37-s − 0.762·43-s + 18/7·49-s − 0.128·61-s − 1.34·67-s + 1.17·73-s − 1.46·79-s + 2.62·91-s − 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6054341601\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6054341601\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| good | 7 | \( 1 + 5 T + p T^{2} \) | 1.7.f |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 5 T + p T^{2} \) | 1.13.f |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 11 T + p T^{2} \) | 1.37.l |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 5 T + p T^{2} \) | 1.43.f |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + T + p T^{2} \) | 1.61.b |
| 67 | \( 1 + 11 T + p T^{2} \) | 1.67.l |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 + 13 T + p T^{2} \) | 1.79.n |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.50895399246613, −14.93082476575570, −14.29734550127955, −13.63430231185630, −13.37560242533110, −12.60925122491814, −12.15075772616263, −11.93191007241623, −10.98753897229376, −10.29313881940118, −9.865058054198126, −9.437383584134301, −9.039675476246710, −8.146675371797415, −7.350152241002656, −7.018659436995729, −6.547884465504351, −5.515840194126939, −5.411890435866045, −4.411718213895592, −3.549787885938542, −3.122959583571416, −2.537763161108414, −1.479989325028149, −0.3053804270379250,
0.3053804270379250, 1.479989325028149, 2.537763161108414, 3.122959583571416, 3.549787885938542, 4.411718213895592, 5.411890435866045, 5.515840194126939, 6.547884465504351, 7.018659436995729, 7.350152241002656, 8.146675371797415, 9.039675476246710, 9.437383584134301, 9.865058054198126, 10.29313881940118, 10.98753897229376, 11.93191007241623, 12.15075772616263, 12.60925122491814, 13.37560242533110, 13.63430231185630, 14.29734550127955, 14.93082476575570, 15.50895399246613
