| L(s) = 1 | + 2·3-s + 4·5-s + 7-s + 9-s + 8·15-s + 2·17-s + 2·19-s + 2·21-s − 8·23-s + 11·25-s − 4·27-s + 2·29-s − 4·31-s + 4·35-s + 6·37-s + 2·41-s − 8·43-s + 4·45-s + 4·47-s + 49-s + 4·51-s + 10·53-s + 4·57-s + 6·59-s + 4·61-s + 63-s − 12·67-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1.78·5-s + 0.377·7-s + 1/3·9-s + 2.06·15-s + 0.485·17-s + 0.458·19-s + 0.436·21-s − 1.66·23-s + 11/5·25-s − 0.769·27-s + 0.371·29-s − 0.718·31-s + 0.676·35-s + 0.986·37-s + 0.312·41-s − 1.21·43-s + 0.596·45-s + 0.583·47-s + 1/7·49-s + 0.560·51-s + 1.37·53-s + 0.529·57-s + 0.781·59-s + 0.512·61-s + 0.125·63-s − 1.46·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54208 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.467787346\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.467787346\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| 11 | \( 1 \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 5 | \( 1 - 4 T + p T^{2} \) | 1.5.ae |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 - 4 T + p T^{2} \) | 1.61.ae |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−14.17841115351796, −14.06679666339436, −13.60726838237474, −13.03864690900018, −12.69280430579115, −11.79073459204037, −11.49629511114798, −10.51430695491183, −10.17163035187253, −9.746348628932635, −9.280985245267189, −8.724927636398715, −8.335332612346531, −7.650774352989741, −7.191780939396432, −6.333638237223574, −5.868925265674911, −5.459670136180798, −4.766185743801000, −3.967472543580647, −3.328658831937676, −2.630352388087268, −2.107234396245941, −1.718490532981195, −0.8053538673000836,
0.8053538673000836, 1.718490532981195, 2.107234396245941, 2.630352388087268, 3.328658831937676, 3.967472543580647, 4.766185743801000, 5.459670136180798, 5.868925265674911, 6.333638237223574, 7.191780939396432, 7.650774352989741, 8.335332612346531, 8.724927636398715, 9.280985245267189, 9.746348628932635, 10.17163035187253, 10.51430695491183, 11.49629511114798, 11.79073459204037, 12.69280430579115, 13.03864690900018, 13.60726838237474, 14.06679666339436, 14.17841115351796
