| L(s) = 1 | + 3-s + 9-s + 2·11-s − 4·13-s − 19-s + 6·23-s − 5·25-s + 27-s − 2·29-s + 4·31-s + 2·33-s − 2·37-s − 4·39-s − 2·41-s − 8·43-s − 8·47-s + 6·53-s − 57-s + 4·59-s + 14·61-s − 2·67-s + 6·69-s + 6·73-s − 5·75-s + 10·79-s + 81-s − 2·87-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s + 0.603·11-s − 1.10·13-s − 0.229·19-s + 1.25·23-s − 25-s + 0.192·27-s − 0.371·29-s + 0.718·31-s + 0.348·33-s − 0.328·37-s − 0.640·39-s − 0.312·41-s − 1.21·43-s − 1.16·47-s + 0.824·53-s − 0.132·57-s + 0.520·59-s + 1.79·61-s − 0.244·67-s + 0.722·69-s + 0.702·73-s − 0.577·75-s + 1.12·79-s + 1/9·81-s − 0.214·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 178752 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 178752 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.729046158\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.729046158\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 19 | \( 1 + T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 18 T + p T^{2} \) | 1.89.s |
| 97 | \( 1 - 16 T + p T^{2} \) | 1.97.aq |
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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
−13.08548928960996, −12.92669350247157, −12.11644581730913, −11.81549875965158, −11.41497320330032, −10.76767136857308, −10.20352788152642, −9.720434153095498, −9.505924141898603, −8.853296520146090, −8.332888483959777, −8.045885793136588, −7.251266554902257, −6.980387685127338, −6.539202006821812, −5.816281134643865, −5.193954481674227, −4.775690689400743, −4.213776026466791, −3.520306839105979, −3.204650452332897, −2.351697702342058, −2.026000955285890, −1.240824918066772, −0.4623605877844208,
0.4623605877844208, 1.240824918066772, 2.026000955285890, 2.351697702342058, 3.204650452332897, 3.520306839105979, 4.213776026466791, 4.775690689400743, 5.193954481674227, 5.816281134643865, 6.539202006821812, 6.980387685127338, 7.251266554902257, 8.045885793136588, 8.332888483959777, 8.853296520146090, 9.505924141898603, 9.720434153095498, 10.20352788152642, 10.76767136857308, 11.41497320330032, 11.81549875965158, 12.11644581730913, 12.92669350247157, 13.08548928960996
