| L(s) = 1 | − 3-s − 5-s + 2·7-s + 9-s − 2·11-s − 2·13-s + 15-s − 2·17-s + 2·19-s − 2·21-s + 2·23-s + 25-s − 27-s − 6·29-s + 4·31-s + 2·33-s − 2·35-s + 2·37-s + 2·39-s − 10·41-s − 8·43-s − 45-s + 2·47-s − 3·49-s + 2·51-s + 6·53-s + 2·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.755·7-s + 1/3·9-s − 0.603·11-s − 0.554·13-s + 0.258·15-s − 0.485·17-s + 0.458·19-s − 0.436·21-s + 0.417·23-s + 1/5·25-s − 0.192·27-s − 1.11·29-s + 0.718·31-s + 0.348·33-s − 0.338·35-s + 0.328·37-s + 0.320·39-s − 1.56·41-s − 1.21·43-s − 0.149·45-s + 0.291·47-s − 3/7·49-s + 0.280·51-s + 0.824·53-s + 0.269·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| good | 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 - 2 T + p T^{2} \) | 1.47.ac |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 2 T + p T^{2} \) | 1.59.ac |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 - 16 T + p T^{2} \) | 1.79.aq |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
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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
−8.705036166814153428105798308798, −7.952538608603195590481697514514, −7.28611523744367416618862524272, −6.50241985693379192242474365828, −5.34080329196659824170043396266, −4.91664731218580831508240373286, −3.95381617705721166703230154549, −2.76503835786649061543002361048, −1.52877886661374831253428270259, 0,
1.52877886661374831253428270259, 2.76503835786649061543002361048, 3.95381617705721166703230154549, 4.91664731218580831508240373286, 5.34080329196659824170043396266, 6.50241985693379192242474365828, 7.28611523744367416618862524272, 7.952538608603195590481697514514, 8.705036166814153428105798308798
