| L(s) = 1 | + 2-s + 4-s − 5-s − 2·7-s + 8-s − 10-s + 2·11-s − 6·13-s − 2·14-s + 16-s − 17-s − 8·19-s − 20-s + 2·22-s + 2·23-s + 25-s − 6·26-s − 2·28-s − 6·29-s − 2·31-s + 32-s − 34-s + 2·35-s + 6·37-s − 8·38-s − 40-s − 2·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.755·7-s + 0.353·8-s − 0.316·10-s + 0.603·11-s − 1.66·13-s − 0.534·14-s + 1/4·16-s − 0.242·17-s − 1.83·19-s − 0.223·20-s + 0.426·22-s + 0.417·23-s + 1/5·25-s − 1.17·26-s − 0.377·28-s − 1.11·29-s − 0.359·31-s + 0.176·32-s − 0.171·34-s + 0.338·35-s + 0.986·37-s − 1.29·38-s − 0.158·40-s − 0.312·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1530 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1530 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 17 | \( 1 + T \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + 14 T + p T^{2} \) | 1.71.o |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 + 14 T + p T^{2} \) | 1.79.o |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
| 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
−9.148818807532985173604938781672, −8.165289990383159221567074208012, −7.17407676622257747637716061141, −6.68589731318848180555968808696, −5.76522650144127058684800715378, −4.69751156295169021925689279438, −4.04695885677905320524362153019, −3.01924876370414985708347922226, −2.02837263408024306182620544902, 0,
2.02837263408024306182620544902, 3.01924876370414985708347922226, 4.04695885677905320524362153019, 4.69751156295169021925689279438, 5.76522650144127058684800715378, 6.68589731318848180555968808696, 7.17407676622257747637716061141, 8.165289990383159221567074208012, 9.148818807532985173604938781672
