| L(s) = 1 | − 7-s − 3·9-s + 4·11-s − 2·13-s + 2·17-s − 4·19-s + 4·23-s + 6·29-s + 31-s − 2·37-s − 6·41-s + 8·47-s + 49-s − 2·53-s − 4·59-s − 2·61-s + 3·63-s + 4·67-s − 16·71-s + 2·73-s − 4·77-s − 16·79-s + 9·81-s + 10·89-s + 2·91-s − 2·97-s − 12·99-s + ⋯ |
| L(s) = 1 | − 0.377·7-s − 9-s + 1.20·11-s − 0.554·13-s + 0.485·17-s − 0.917·19-s + 0.834·23-s + 1.11·29-s + 0.179·31-s − 0.328·37-s − 0.937·41-s + 1.16·47-s + 1/7·49-s − 0.274·53-s − 0.520·59-s − 0.256·61-s + 0.377·63-s + 0.488·67-s − 1.89·71-s + 0.234·73-s − 0.455·77-s − 1.80·79-s + 81-s + 1.05·89-s + 0.209·91-s − 0.203·97-s − 1.20·99-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 86800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 86800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 31 | \( 1 - T \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 16 T + p T^{2} \) | 1.71.q |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + 16 T + p T^{2} \) | 1.79.q |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−14.15674012144510, −13.76116347221786, −13.21669747993548, −12.47738258013964, −12.25037239940350, −11.68002646318465, −11.33119411287793, −10.57360979506323, −10.27602889735247, −9.593201059958707, −9.100736455451526, −8.613984893981211, −8.340679558908429, −7.418598844866932, −7.069173282863963, −6.333791125828689, −6.120699698343556, −5.395616945300591, −4.764694133132286, −4.255441042022396, −3.509086086191365, −3.015455750605730, −2.447315920325803, −1.618112532887956, −0.8547819829127071, 0,
0.8547819829127071, 1.618112532887956, 2.447315920325803, 3.015455750605730, 3.509086086191365, 4.255441042022396, 4.764694133132286, 5.395616945300591, 6.120699698343556, 6.333791125828689, 7.069173282863963, 7.418598844866932, 8.340679558908429, 8.613984893981211, 9.100736455451526, 9.593201059958707, 10.27602889735247, 10.57360979506323, 11.33119411287793, 11.68002646318465, 12.25037239940350, 12.47738258013964, 13.21669747993548, 13.76116347221786, 14.15674012144510
