| L(s) = 1 | − 2-s + 2·3-s + 4-s − 2·6-s − 8-s + 9-s + 4·11-s + 2·12-s + 2·13-s + 16-s − 8·17-s − 18-s + 5·19-s − 4·22-s − 23-s − 2·24-s − 2·26-s − 4·27-s − 10·29-s + 4·31-s − 32-s + 8·33-s + 8·34-s + 36-s − 5·38-s + 4·39-s + 7·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.15·3-s + 1/2·4-s − 0.816·6-s − 0.353·8-s + 1/3·9-s + 1.20·11-s + 0.577·12-s + 0.554·13-s + 1/4·16-s − 1.94·17-s − 0.235·18-s + 1.14·19-s − 0.852·22-s − 0.208·23-s − 0.408·24-s − 0.392·26-s − 0.769·27-s − 1.85·29-s + 0.718·31-s − 0.176·32-s + 1.39·33-s + 1.37·34-s + 1/6·36-s − 0.811·38-s + 0.640·39-s + 1.09·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 68450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 68450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.814487644\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.814487644\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 37 | \( 1 \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 8 T + p T^{2} \) | 1.17.i |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 23 | \( 1 + T + p T^{2} \) | 1.23.b |
| 29 | \( 1 + 10 T + p T^{2} \) | 1.29.k |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 41 | \( 1 - 7 T + p T^{2} \) | 1.41.ah |
| 43 | \( 1 - 9 T + p T^{2} \) | 1.43.aj |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 + 3 T + p T^{2} \) | 1.53.d |
| 59 | \( 1 - 11 T + p T^{2} \) | 1.59.al |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 - 14 T + p T^{2} \) | 1.71.ao |
| 73 | \( 1 - 3 T + p T^{2} \) | 1.73.ad |
| 79 | \( 1 - 11 T + p T^{2} \) | 1.79.al |
| 83 | \( 1 - 8 T + p T^{2} \) | 1.83.ai |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 97 | \( 1 - 8 T + p T^{2} \) | 1.97.ai |
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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
−14.11763982331195, −13.79806099275460, −13.28480125572326, −12.75311712656066, −12.08350934549994, −11.46026503067765, −11.09895924468019, −10.77109541921374, −9.670147672233862, −9.488106643557917, −9.081016000717442, −8.686625001297551, −8.073754329439804, −7.617612495218471, −7.045598990129348, −6.486537424659413, −5.983195773810975, −5.269483966065654, −4.330465783325616, −3.822956804428704, −3.415239316507000, −2.463722764475810, −2.178996736043445, −1.381921253229553, −0.5929292731353445,
0.5929292731353445, 1.381921253229553, 2.178996736043445, 2.463722764475810, 3.415239316507000, 3.822956804428704, 4.330465783325616, 5.269483966065654, 5.983195773810975, 6.486537424659413, 7.045598990129348, 7.617612495218471, 8.073754329439804, 8.686625001297551, 9.081016000717442, 9.488106643557917, 9.670147672233862, 10.77109541921374, 11.09895924468019, 11.46026503067765, 12.08350934549994, 12.75311712656066, 13.28480125572326, 13.79806099275460, 14.11763982331195
