| L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s + 8-s + 9-s + 10-s + 4·11-s + 12-s + 13-s + 15-s + 16-s + 18-s + 6·19-s + 20-s + 4·22-s + 24-s + 25-s + 26-s + 27-s + 6·29-s + 30-s + 10·31-s + 32-s + 4·33-s + 36-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 + 1.20·11-s + 0.288·12-s + 0.277·13-s + 0.258·15-s + 1/4·16-s + 0.235·18-s + 1.37·19-s + 0.223·20-s + 0.852·22-s + 0.204·24-s + 1/5·25-s + 0.196·26-s + 0.192·27-s + 1.11·29-s + 0.182·30-s + 1.79·31-s + 0.176·32-s + 0.696·33-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.582925963\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.582925963\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| good | 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 10 T + p T^{2} \) | 1.31.ak |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 10 T + p T^{2} \) | 1.83.ak |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.71390392556630, −15.03725644490354, −14.54201027821756, −13.92049087598744, −13.68254224908394, −13.29925070647763, −12.25173279467699, −12.03082262989267, −11.56412019730398, −10.67288817918552, −10.11427354789103, −9.625791480106148, −8.947885509764238, −8.406975477950199, −7.755901757572732, −6.911456573578659, −6.566709901080448, −5.952022457912155, −5.040895139526846, −4.666152713809556, −3.699801638639734, −3.291832828720908, −2.549076516710022, −1.613448793082832, −1.022005234744641,
1.022005234744641, 1.613448793082832, 2.549076516710022, 3.291832828720908, 3.699801638639734, 4.666152713809556, 5.040895139526846, 5.952022457912155, 6.566709901080448, 6.911456573578659, 7.755901757572732, 8.406975477950199, 8.947885509764238, 9.625791480106148, 10.11427354789103, 10.67288817918552, 11.56412019730398, 12.03082262989267, 12.25173279467699, 13.29925070647763, 13.68254224908394, 13.92049087598744, 14.54201027821756, 15.03725644490354, 15.71390392556630
