| L(s) = 1 | + 2-s + 3-s − 4-s + 6-s − 3·8-s + 9-s − 4·11-s − 12-s − 2·13-s − 16-s + 2·17-s + 18-s + 19-s − 4·22-s − 3·24-s − 2·26-s + 27-s + 6·29-s − 8·31-s + 5·32-s − 4·33-s + 2·34-s − 36-s − 6·37-s + 38-s − 2·39-s − 10·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.408·6-s − 1.06·8-s + 1/3·9-s − 1.20·11-s − 0.288·12-s − 0.554·13-s − 1/4·16-s + 0.485·17-s + 0.235·18-s + 0.229·19-s − 0.852·22-s − 0.612·24-s − 0.392·26-s + 0.192·27-s + 1.11·29-s − 1.43·31-s + 0.883·32-s − 0.696·33-s + 0.342·34-s − 1/6·36-s − 0.986·37-s + 0.162·38-s − 0.320·39-s − 1.56·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.567009575\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.567009575\) |
| \(L(\frac{3}{2})\) |
|
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 | 3 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 \) | |
| 19 | \( 1 - T \) | |
| good | 2 | \( 1 - T + p T^{2} \) | 1.2.ab |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−14.11993712875271, −13.69194916871019, −13.14041271335659, −12.80031610852548, −12.31273568911666, −11.88085623391722, −11.17919347362989, −10.49451543249092, −10.04692635818395, −9.635914154390605, −8.996144894897017, −8.523425091691987, −7.970860184780715, −7.575424764784231, −6.820970656735438, −6.321411879288690, −5.427392598708249, −5.120142004751244, −4.786427247403780, −3.845043747060784, −3.480001566777247, −2.852249969613396, −2.310047175014510, −1.439657559532230, −0.3506736341372644,
0.3506736341372644, 1.439657559532230, 2.310047175014510, 2.852249969613396, 3.480001566777247, 3.845043747060784, 4.786427247403780, 5.120142004751244, 5.427392598708249, 6.321411879288690, 6.820970656735438, 7.575424764784231, 7.970860184780715, 8.523425091691987, 8.996144894897017, 9.635914154390605, 10.04692635818395, 10.49451543249092, 11.17919347362989, 11.88085623391722, 12.31273568911666, 12.80031610852548, 13.14041271335659, 13.69194916871019, 14.11993712875271
