| L(s) = 1 | − 3-s − 2·5-s + 2·7-s + 9-s − 11-s − 2·13-s + 2·15-s + 2·19-s − 2·21-s − 25-s − 27-s − 8·29-s + 4·31-s + 33-s − 4·35-s − 6·37-s + 2·39-s − 4·41-s + 6·43-s − 2·45-s − 8·47-s − 3·49-s − 6·53-s + 2·55-s − 2·57-s − 4·59-s − 6·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s + 0.755·7-s + 1/3·9-s − 0.301·11-s − 0.554·13-s + 0.516·15-s + 0.458·19-s − 0.436·21-s − 1/5·25-s − 0.192·27-s − 1.48·29-s + 0.718·31-s + 0.174·33-s − 0.676·35-s − 0.986·37-s + 0.320·39-s − 0.624·41-s + 0.914·43-s − 0.298·45-s − 1.16·47-s − 3/7·49-s − 0.824·53-s + 0.269·55-s − 0.264·57-s − 0.520·59-s − 0.768·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 11 | \( 1 + T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 8 T + p T^{2} \) | 1.29.i |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + 4 T + p T^{2} \) | 1.41.e |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + 14 T + p T^{2} \) | 1.79.o |
| 83 | \( 1 + 8 T + p T^{2} \) | 1.83.i |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−9.587167765061418454974098068595, −8.523279688734374245273759606496, −7.69466396216281512547117145758, −7.20477175532301789858388521427, −5.98682525365222690906249602329, −5.05832534418089349774989826260, −4.34993830639755531513932353698, −3.22476990096031886640156983037, −1.69334135129200147215330834471, 0,
1.69334135129200147215330834471, 3.22476990096031886640156983037, 4.34993830639755531513932353698, 5.05832534418089349774989826260, 5.98682525365222690906249602329, 7.20477175532301789858388521427, 7.69466396216281512547117145758, 8.523279688734374245273759606496, 9.587167765061418454974098068595
