| L(s) = 1 | + 2·3-s − 5-s − 4·7-s + 9-s + 6·11-s − 13-s − 2·15-s − 6·17-s − 2·19-s − 8·21-s + 6·23-s + 25-s − 4·27-s + 6·29-s + 2·31-s + 12·33-s + 4·35-s − 2·37-s − 2·39-s − 6·41-s − 2·43-s − 45-s − 12·47-s + 9·49-s − 12·51-s − 6·53-s − 6·55-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.447·5-s − 1.51·7-s + 1/3·9-s + 1.80·11-s − 0.277·13-s − 0.516·15-s − 1.45·17-s − 0.458·19-s − 1.74·21-s + 1.25·23-s + 1/5·25-s − 0.769·27-s + 1.11·29-s + 0.359·31-s + 2.08·33-s + 0.676·35-s − 0.328·37-s − 0.320·39-s − 0.937·41-s − 0.304·43-s − 0.149·45-s − 1.75·47-s + 9/7·49-s − 1.68·51-s − 0.824·53-s − 0.809·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 13 | \( 1 + T \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−8.385892994426974387064320936714, −7.15031833255794122549807157204, −6.67646862518206244633880985882, −6.22915729835157443087981415258, −4.75317284270628420700819478225, −4.02757025612801301625618344495, −3.26724143545532321208050075563, −2.78007888158177502434541513722, −1.55547954514278666952931744266, 0,
1.55547954514278666952931744266, 2.78007888158177502434541513722, 3.26724143545532321208050075563, 4.02757025612801301625618344495, 4.75317284270628420700819478225, 6.22915729835157443087981415258, 6.67646862518206244633880985882, 7.15031833255794122549807157204, 8.385892994426974387064320936714
