| L(s) = 1 | − 3-s + 5-s + 9-s − 11-s − 2·13-s − 15-s − 6·17-s − 4·19-s + 25-s − 27-s + 6·29-s − 8·31-s + 33-s + 6·37-s + 2·39-s + 10·41-s − 4·43-s + 45-s − 8·47-s − 7·49-s + 6·51-s − 10·53-s − 55-s + 4·57-s − 12·59-s + 6·61-s − 2·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1/3·9-s − 0.301·11-s − 0.554·13-s − 0.258·15-s − 1.45·17-s − 0.917·19-s + 1/5·25-s − 0.192·27-s + 1.11·29-s − 1.43·31-s + 0.174·33-s + 0.986·37-s + 0.320·39-s + 1.56·41-s − 0.609·43-s + 0.149·45-s − 1.16·47-s − 49-s + 0.840·51-s − 1.37·53-s − 0.134·55-s + 0.529·57-s − 1.56·59-s + 0.768·61-s − 0.248·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 3 | \( 1 + T \) | |
| 5 | \( 1 - T \) | |
| 11 | \( 1 + T \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 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 |
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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
−9.333630071934027258696545399878, −8.477833861211703602688168344638, −7.50964419679360078093821068413, −6.58507099264904515968522179448, −6.02403827806525130766769933380, −4.91554972608167666796589856516, −4.31998102966678622348392629220, −2.82907835433343182491059338466, −1.77349882244994403788180956111, 0,
1.77349882244994403788180956111, 2.82907835433343182491059338466, 4.31998102966678622348392629220, 4.91554972608167666796589856516, 6.02403827806525130766769933380, 6.58507099264904515968522179448, 7.50964419679360078093821068413, 8.477833861211703602688168344638, 9.333630071934027258696545399878
