| L(s) = 1 | − 2-s + 3-s + 4-s − 2·5-s − 6-s − 7-s − 8-s + 9-s + 2·10-s + 12-s + 13-s + 14-s − 2·15-s + 16-s + 2·17-s − 18-s + 4·19-s − 2·20-s − 21-s − 4·23-s − 24-s − 25-s − 26-s + 27-s − 28-s + 6·29-s + 2·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.894·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.632·10-s + 0.288·12-s + 0.277·13-s + 0.267·14-s − 0.516·15-s + 1/4·16-s + 0.485·17-s − 0.235·18-s + 0.917·19-s − 0.447·20-s − 0.218·21-s − 0.834·23-s − 0.204·24-s − 1/5·25-s − 0.196·26-s + 0.192·27-s − 0.188·28-s + 1.11·29-s + 0.365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 66066 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 66066 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.387261422\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.387261422\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 7 | \( 1 + T \) | |
| 11 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 16 T + p T^{2} \) | 1.71.aq |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 8 T + p T^{2} \) | 1.83.i |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
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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
−14.14839826406090, −13.84200711874280, −13.28465830319953, −12.36907309728343, −12.22400694124620, −11.80238566355526, −11.02981440003133, −10.66480936626103, −9.978935418474567, −9.641047037530799, −9.074096038497997, −8.460716816942459, −8.072289501515319, −7.626916437629066, −7.116594248517508, −6.559786786051399, −5.914037269466991, −5.251069017363247, −4.496952474125015, −3.770006197156461, −3.402905934144917, −2.784566055475967, −2.011945988104188, −1.227741510808887, −0.4602156792126370,
0.4602156792126370, 1.227741510808887, 2.011945988104188, 2.784566055475967, 3.402905934144917, 3.770006197156461, 4.496952474125015, 5.251069017363247, 5.914037269466991, 6.559786786051399, 7.116594248517508, 7.626916437629066, 8.072289501515319, 8.460716816942459, 9.074096038497997, 9.641047037530799, 9.978935418474567, 10.66480936626103, 11.02981440003133, 11.80238566355526, 12.22400694124620, 12.36907309728343, 13.28465830319953, 13.84200711874280, 14.14839826406090
