| L(s) = 1 | + 5-s − 3·7-s − 13-s − 2·17-s + 19-s − 6·23-s − 4·25-s + 10·29-s + 4·31-s − 3·35-s + 11·37-s + 3·41-s + 10·43-s − 6·47-s + 2·49-s + 2·53-s + 14·59-s + 4·61-s − 65-s − 12·67-s − 15·71-s + 3·73-s − 14·79-s − 17·83-s − 2·85-s − 3·89-s + 3·91-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.13·7-s − 0.277·13-s − 0.485·17-s + 0.229·19-s − 1.25·23-s − 4/5·25-s + 1.85·29-s + 0.718·31-s − 0.507·35-s + 1.80·37-s + 0.468·41-s + 1.52·43-s − 0.875·47-s + 2/7·49-s + 0.274·53-s + 1.82·59-s + 0.512·61-s − 0.124·65-s − 1.46·67-s − 1.78·71-s + 0.351·73-s − 1.57·79-s − 1.86·83-s − 0.216·85-s − 0.317·89-s + 0.314·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8208 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 19 | \( 1 - T \) | |
| good | 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 11 T + p T^{2} \) | 1.37.al |
| 41 | \( 1 - 3 T + p T^{2} \) | 1.41.ad |
| 43 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 - 14 T + p T^{2} \) | 1.59.ao |
| 61 | \( 1 - 4 T + p T^{2} \) | 1.61.ae |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + 15 T + p T^{2} \) | 1.71.p |
| 73 | \( 1 - 3 T + p T^{2} \) | 1.73.ad |
| 79 | \( 1 + 14 T + p T^{2} \) | 1.79.o |
| 83 | \( 1 + 17 T + p T^{2} \) | 1.83.r |
| 89 | \( 1 + 3 T + p T^{2} \) | 1.89.d |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−7.42226286085077097363556254100, −6.64784415003229959179361247689, −6.09899789343223961104905847859, −5.63413077128856428108660040475, −4.46724297402061547886571837061, −4.03235000279299920147547976247, −2.83704938023478593999842614918, −2.49302392983584359950509482209, −1.20089483873441336729736932597, 0,
1.20089483873441336729736932597, 2.49302392983584359950509482209, 2.83704938023478593999842614918, 4.03235000279299920147547976247, 4.46724297402061547886571837061, 5.63413077128856428108660040475, 6.09899789343223961104905847859, 6.64784415003229959179361247689, 7.42226286085077097363556254100
