| L(s) = 1 | + 2·3-s + 5-s − 2·7-s + 9-s − 4·11-s + 2·13-s + 2·15-s + 4·17-s − 4·19-s − 4·21-s + 4·23-s + 25-s − 4·27-s − 2·29-s − 8·31-s − 8·33-s − 2·35-s − 4·37-s + 4·39-s − 2·41-s + 2·43-s + 45-s − 6·47-s − 3·49-s + 8·51-s − 4·55-s − 8·57-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.447·5-s − 0.755·7-s + 1/3·9-s − 1.20·11-s + 0.554·13-s + 0.516·15-s + 0.970·17-s − 0.917·19-s − 0.872·21-s + 0.834·23-s + 1/5·25-s − 0.769·27-s − 0.371·29-s − 1.43·31-s − 1.39·33-s − 0.338·35-s − 0.657·37-s + 0.640·39-s − 0.312·41-s + 0.304·43-s + 0.149·45-s − 0.875·47-s − 3/7·49-s + 1.12·51-s − 0.539·55-s − 1.05·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 79 | \( 1 - T \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + 16 T + p T^{2} \) | 1.71.q |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 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
−7.67008255477933760991433208866, −7.22329918228592839586944975148, −6.20612396130760631485655250596, −5.62592885542084982816934982754, −4.82195068433520528627795320486, −3.63479557909904202428650762890, −3.20589700370284364908510613284, −2.45528871816985241258107229214, −1.59422500695158483214670044278, 0,
1.59422500695158483214670044278, 2.45528871816985241258107229214, 3.20589700370284364908510613284, 3.63479557909904202428650762890, 4.82195068433520528627795320486, 5.62592885542084982816934982754, 6.20612396130760631485655250596, 7.22329918228592839586944975148, 7.67008255477933760991433208866
