| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 8-s + 9-s − 11-s + 12-s + 2·13-s + 16-s + 4·17-s − 18-s − 6·19-s + 22-s − 4·23-s − 24-s − 5·25-s − 2·26-s + 27-s − 10·29-s − 6·31-s − 32-s − 33-s − 4·34-s + 36-s − 6·37-s + 6·38-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.301·11-s + 0.288·12-s + 0.554·13-s + 1/4·16-s + 0.970·17-s − 0.235·18-s − 1.37·19-s + 0.213·22-s − 0.834·23-s − 0.204·24-s − 25-s − 0.392·26-s + 0.192·27-s − 1.85·29-s − 1.07·31-s − 0.176·32-s − 0.174·33-s − 0.685·34-s + 1/6·36-s − 0.986·37-s + 0.973·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 10 T + p T^{2} \) | 1.29.k |
| 31 | \( 1 + 6 T + p T^{2} \) | 1.31.g |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 - 12 T + p T^{2} \) | 1.41.am |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 12 T + p T^{2} \) | 1.73.am |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 14 T + p T^{2} \) | 1.83.o |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−8.291395379285966933177433923359, −7.71587761736138043646912415143, −7.05711515857633022078951690658, −6.04092748710153501284017966573, −5.46410587935044694748407940394, −4.07422394736781959429805037262, −3.52231105656348010954914510103, −2.30952952489411018960352501765, −1.60096048597049773160721959429, 0,
1.60096048597049773160721959429, 2.30952952489411018960352501765, 3.52231105656348010954914510103, 4.07422394736781959429805037262, 5.46410587935044694748407940394, 6.04092748710153501284017966573, 7.05711515857633022078951690658, 7.71587761736138043646912415143, 8.291395379285966933177433923359
