| L(s) = 1 | − 3-s − 7-s + 9-s − 2·13-s − 4·17-s + 4·19-s + 21-s + 6·23-s − 5·25-s − 27-s − 2·29-s + 6·37-s + 2·39-s − 8·41-s + 8·43-s + 4·47-s + 49-s + 4·51-s + 6·53-s − 4·57-s − 14·61-s − 63-s + 4·67-s − 6·69-s + 2·71-s + 2·73-s + 5·75-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.377·7-s + 1/3·9-s − 0.554·13-s − 0.970·17-s + 0.917·19-s + 0.218·21-s + 1.25·23-s − 25-s − 0.192·27-s − 0.371·29-s + 0.986·37-s + 0.320·39-s − 1.24·41-s + 1.21·43-s + 0.583·47-s + 1/7·49-s + 0.560·51-s + 0.824·53-s − 0.529·57-s − 1.79·61-s − 0.125·63-s + 0.488·67-s − 0.722·69-s + 0.237·71-s + 0.234·73-s + 0.577·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.302756154\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.302756154\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 11 | \( 1 \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 8 T + p T^{2} \) | 1.41.i |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 2 T + p T^{2} \) | 1.71.ac |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
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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
−13.32153535942087, −12.71910199949482, −12.34012215216041, −11.79135949896189, −11.35340715333555, −11.01433642016039, −10.39145284887776, −9.970518986831125, −9.403622811272173, −9.093340921989963, −8.546493122106668, −7.706999174698325, −7.439142864821214, −6.943590587682851, −6.361695428742539, −5.921363574949223, −5.322122033384429, −4.902003552325201, −4.291587270785317, −3.782873196758185, −3.062187248288956, −2.527949575274483, −1.859330828995490, −1.066712391562557, −0.3846915858540501,
0.3846915858540501, 1.066712391562557, 1.859330828995490, 2.527949575274483, 3.062187248288956, 3.782873196758185, 4.291587270785317, 4.902003552325201, 5.322122033384429, 5.921363574949223, 6.361695428742539, 6.943590587682851, 7.439142864821214, 7.706999174698325, 8.546493122106668, 9.093340921989963, 9.403622811272173, 9.970518986831125, 10.39145284887776, 11.01433642016039, 11.35340715333555, 11.79135949896189, 12.34012215216041, 12.71910199949482, 13.32153535942087
