| L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s + 8-s + 9-s + 10-s + 6·11-s + 12-s + 15-s + 16-s + 6·17-s + 18-s − 6·19-s + 20-s + 6·22-s + 2·23-s + 24-s + 25-s + 27-s − 6·29-s + 30-s − 4·31-s + 32-s + 6·33-s + 6·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s + 0.353·8-s + 1/3·9-s + 0.316·10-s + 1.80·11-s + 0.288·12-s + 0.258·15-s + 1/4·16-s + 1.45·17-s + 0.235·18-s − 1.37·19-s + 0.223·20-s + 1.27·22-s + 0.417·23-s + 0.204·24-s + 1/5·25-s + 0.192·27-s − 1.11·29-s + 0.182·30-s − 0.718·31-s + 0.176·32-s + 1.04·33-s + 1.02·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 248430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 248430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.702776405\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.702776405\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 3 | \( 1 - T \) | |
| 5 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 - 8 T + p T^{2} \) | 1.41.ai |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 4 T + p T^{2} \) | 1.71.ae |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 16 T + p T^{2} \) | 1.83.aq |
| 89 | \( 1 + 12 T + p T^{2} \) | 1.89.m |
| 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
−12.88710419894398, −12.62836410585731, −11.91896597557931, −11.56634497168753, −11.17879475993785, −10.52281008706011, −10.06045527535256, −9.511600889162665, −9.251310052045489, −8.677973940744980, −8.147215921814927, −7.671272652773353, −7.034987252273015, −6.635593916978400, −6.224415709382468, −5.682614237823264, −5.182055458742461, −4.511103540421114, −4.016494866402556, −3.570081455425972, −3.200326375602202, −2.377489916799843, −1.852094979225714, −1.394635804111774, −0.6805949847320348,
0.6805949847320348, 1.394635804111774, 1.852094979225714, 2.377489916799843, 3.200326375602202, 3.570081455425972, 4.016494866402556, 4.511103540421114, 5.182055458742461, 5.682614237823264, 6.224415709382468, 6.635593916978400, 7.034987252273015, 7.671272652773353, 8.147215921814927, 8.677973940744980, 9.251310052045489, 9.511600889162665, 10.06045527535256, 10.52281008706011, 11.17879475993785, 11.56634497168753, 11.91896597557931, 12.62836410585731, 12.88710419894398
