| L(s) = 1 | + 2·3-s + 5-s − 4·7-s + 9-s + 4·11-s + 2·15-s − 8·17-s + 4·19-s − 8·21-s − 2·23-s + 25-s − 4·27-s + 6·29-s − 31-s + 8·33-s − 4·35-s + 4·37-s − 6·41-s − 6·43-s + 45-s − 8·47-s + 9·49-s − 16·51-s + 12·53-s + 4·55-s + 8·57-s − 4·59-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.447·5-s − 1.51·7-s + 1/3·9-s + 1.20·11-s + 0.516·15-s − 1.94·17-s + 0.917·19-s − 1.74·21-s − 0.417·23-s + 1/5·25-s − 0.769·27-s + 1.11·29-s − 0.179·31-s + 1.39·33-s − 0.676·35-s + 0.657·37-s − 0.937·41-s − 0.914·43-s + 0.149·45-s − 1.16·47-s + 9/7·49-s − 2.24·51-s + 1.64·53-s + 0.539·55-s + 1.05·57-s − 0.520·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 31 | \( 1 + T \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 + 8 T + p T^{2} \) | 1.17.i |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 2 T + p T^{2} \) | 1.83.ac |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 + 18 T + p T^{2} \) | 1.97.s |
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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
−7.19855422477612201948548702665, −6.51064447517598676788754306998, −6.37222067633058161380817766785, −5.28257846977330367584587441679, −4.27016584088156394931593314622, −3.66475309536403829068260025084, −2.97064147358241148912227417411, −2.38790810230074948659508553086, −1.42247479066643426534898941044, 0,
1.42247479066643426534898941044, 2.38790810230074948659508553086, 2.97064147358241148912227417411, 3.66475309536403829068260025084, 4.27016584088156394931593314622, 5.28257846977330367584587441679, 6.37222067633058161380817766785, 6.51064447517598676788754306998, 7.19855422477612201948548702665
