| L(s) = 1 | − 3-s − 2·5-s + 9-s + 11-s + 2·13-s + 2·15-s + 6·17-s − 4·19-s + 4·23-s − 25-s − 27-s − 6·29-s − 4·31-s − 33-s + 37-s − 2·39-s − 6·41-s − 4·43-s − 2·45-s + 8·47-s − 7·49-s − 6·51-s − 10·53-s − 2·55-s + 4·57-s + 2·61-s − 4·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s + 1/3·9-s + 0.301·11-s + 0.554·13-s + 0.516·15-s + 1.45·17-s − 0.917·19-s + 0.834·23-s − 1/5·25-s − 0.192·27-s − 1.11·29-s − 0.718·31-s − 0.174·33-s + 0.164·37-s − 0.320·39-s − 0.937·41-s − 0.609·43-s − 0.298·45-s + 1.16·47-s − 49-s − 0.840·51-s − 1.37·53-s − 0.269·55-s + 0.529·57-s + 0.256·61-s − 0.496·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9768 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 37 | \( 1 - T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−7.36351445553741088290978323292, −6.65865299612232669223662535364, −5.97244158244895265244965653429, −5.28565840331855219923861852438, −4.58640382836013945008296894204, −3.65344962138666470403302250013, −3.41654814639989437401056155637, −2.01735661039112930056099486941, −1.07526534049776055484248113288, 0,
1.07526534049776055484248113288, 2.01735661039112930056099486941, 3.41654814639989437401056155637, 3.65344962138666470403302250013, 4.58640382836013945008296894204, 5.28565840331855219923861852438, 5.97244158244895265244965653429, 6.65865299612232669223662535364, 7.36351445553741088290978323292
