| L(s) = 1 | − 3-s − 5-s − 3·7-s − 2·9-s − 4·13-s + 15-s + 3·17-s + 3·19-s + 3·21-s − 4·23-s + 25-s + 5·27-s − 29-s + 5·31-s + 3·35-s − 5·37-s + 4·39-s − 2·41-s + 8·43-s + 2·45-s + 10·47-s + 2·49-s − 3·51-s + 11·53-s − 3·57-s + 61-s + 6·63-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 1.13·7-s − 2/3·9-s − 1.10·13-s + 0.258·15-s + 0.727·17-s + 0.688·19-s + 0.654·21-s − 0.834·23-s + 1/5·25-s + 0.962·27-s − 0.185·29-s + 0.898·31-s + 0.507·35-s − 0.821·37-s + 0.640·39-s − 0.312·41-s + 1.21·43-s + 0.298·45-s + 1.45·47-s + 2/7·49-s − 0.420·51-s + 1.51·53-s − 0.397·57-s + 0.128·61-s + 0.755·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 11 | \( 1 \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 - 3 T + p T^{2} \) | 1.19.ad |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + T + p T^{2} \) | 1.29.b |
| 31 | \( 1 - 5 T + p T^{2} \) | 1.31.af |
| 37 | \( 1 + 5 T + p T^{2} \) | 1.37.f |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 - 10 T + p T^{2} \) | 1.47.ak |
| 53 | \( 1 - 11 T + p T^{2} \) | 1.53.al |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - T + p T^{2} \) | 1.61.ab |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 5 T + p T^{2} \) | 1.71.f |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 15 T + p T^{2} \) | 1.89.p |
| 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
−7.22005454224532431550352435242, −6.72599763192045535035949604808, −5.80099131667937013708300149271, −5.52293790034530269118774936371, −4.60011495907793837450689779245, −3.77021167881013715222576259875, −3.02612080744953847023533598033, −2.38211418563783881754853929774, −0.869698312892042500697107165018, 0,
0.869698312892042500697107165018, 2.38211418563783881754853929774, 3.02612080744953847023533598033, 3.77021167881013715222576259875, 4.60011495907793837450689779245, 5.52293790034530269118774936371, 5.80099131667937013708300149271, 6.72599763192045535035949604808, 7.22005454224532431550352435242
