| L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s + 4·7-s + 8-s + 9-s + 10-s − 12-s + 2·13-s + 4·14-s − 15-s + 16-s + 2·17-s + 18-s + 8·19-s + 20-s − 4·21-s − 24-s + 25-s + 2·26-s − 27-s + 4·28-s − 2·29-s − 30-s − 8·31-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 1.51·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.288·12-s + 0.554·13-s + 1.06·14-s − 0.258·15-s + 1/4·16-s + 0.485·17-s + 0.235·18-s + 1.83·19-s + 0.223·20-s − 0.872·21-s − 0.204·24-s + 1/5·25-s + 0.392·26-s − 0.192·27-s + 0.755·28-s − 0.371·29-s − 0.182·30-s − 1.43·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.640177200\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.640177200\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| good | 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 14 T + p T^{2} \) | 1.53.ao |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−8.479883250955093266895744192295, −7.40696925033699638160691249365, −7.27752953900878357828554051261, −5.88014804790306627418533622087, −5.55307889866064846829275639149, −4.92089940314369655781504062688, −4.06246632062950272760150417569, −3.12548788757864162693227054094, −1.85410703674039090636159152963, −1.17775005180092691264201056199,
1.17775005180092691264201056199, 1.85410703674039090636159152963, 3.12548788757864162693227054094, 4.06246632062950272760150417569, 4.92089940314369655781504062688, 5.55307889866064846829275639149, 5.88014804790306627418533622087, 7.27752953900878357828554051261, 7.40696925033699638160691249365, 8.479883250955093266895744192295
