| L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s + 3·7-s + 8-s + 9-s + 10-s − 11-s + 12-s + 3·13-s + 3·14-s + 15-s + 16-s − 6·17-s + 18-s − 4·19-s + 20-s + 3·21-s − 22-s − 23-s + 24-s + 25-s + 3·26-s + 27-s + 3·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s + 1.13·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.301·11-s + 0.288·12-s + 0.832·13-s + 0.801·14-s + 0.258·15-s + 1/4·16-s − 1.45·17-s + 0.235·18-s − 0.917·19-s + 0.223·20-s + 0.654·21-s − 0.213·22-s − 0.208·23-s + 0.204·24-s + 1/5·25-s + 0.588·26-s + 0.192·27-s + 0.566·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.599150063\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.599150063\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 11 | \( 1 + T \) | |
| 89 | \( 1 + T \) | |
| good | 7 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 13 | \( 1 - 3 T + p T^{2} \) | 1.13.ad |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + T + p T^{2} \) | 1.23.b |
| 29 | \( 1 - 5 T + p T^{2} \) | 1.29.af |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 9 T + p T^{2} \) | 1.59.aj |
| 61 | \( 1 - 4 T + p T^{2} \) | 1.61.ae |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 + 13 T + p T^{2} \) | 1.71.n |
| 73 | \( 1 - 13 T + p T^{2} \) | 1.73.an |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + 3 T + p T^{2} \) | 1.83.d |
| 97 | \( 1 + 4 T + p T^{2} \) | 1.97.e |
| 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
−15.23443485870808, −14.35418494901121, −14.18562533623896, −13.69514814335372, −13.00484742392984, −12.85580059073221, −11.93776508254319, −11.45451908322747, −10.89198682638997, −10.46362878160405, −9.901410011915081, −8.922742824494954, −8.601244310098631, −8.150549309254588, −7.455455413965207, −6.675585315075798, −6.323051824556971, −5.567351545621214, −4.845368777188973, −4.375663835504230, −3.889642877626178, −2.879376798979606, −2.301608549230481, −1.785488124573615, −0.8634991829539185,
0.8634991829539185, 1.785488124573615, 2.301608549230481, 2.879376798979606, 3.889642877626178, 4.375663835504230, 4.845368777188973, 5.567351545621214, 6.323051824556971, 6.675585315075798, 7.455455413965207, 8.150549309254588, 8.601244310098631, 8.922742824494954, 9.901410011915081, 10.46362878160405, 10.89198682638997, 11.45451908322747, 11.93776508254319, 12.85580059073221, 13.00484742392984, 13.69514814335372, 14.18562533623896, 14.35418494901121, 15.23443485870808
