| L(s) = 1 | − 5-s − 2·11-s + 4·13-s + 25-s − 2·29-s − 10·31-s + 4·37-s − 6·41-s + 4·43-s − 7·49-s − 2·53-s + 2·55-s − 6·59-s + 6·61-s − 4·65-s + 4·67-s + 8·71-s + 14·73-s − 2·79-s − 4·83-s − 14·89-s − 6·97-s + 18·101-s + 4·103-s − 16·107-s + 10·109-s − 8·113-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.603·11-s + 1.10·13-s + 1/5·25-s − 0.371·29-s − 1.79·31-s + 0.657·37-s − 0.937·41-s + 0.609·43-s − 49-s − 0.274·53-s + 0.269·55-s − 0.781·59-s + 0.768·61-s − 0.496·65-s + 0.488·67-s + 0.949·71-s + 1.63·73-s − 0.225·79-s − 0.439·83-s − 1.48·89-s − 0.609·97-s + 1.79·101-s + 0.394·103-s − 1.54·107-s + 0.957·109-s − 0.752·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + 2 T + p T^{2} \) | 1.79.c |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 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
−7.84854250279194552058818388017, −7.10144076013417515588056530153, −6.35325928031999033219085293354, −5.57893809623021291703244601007, −4.91469175444992974962636300619, −3.90550565366817518772456567813, −3.41429344993515331662632879524, −2.34998712360627625599414754082, −1.30588355911914939053038801182, 0,
1.30588355911914939053038801182, 2.34998712360627625599414754082, 3.41429344993515331662632879524, 3.90550565366817518772456567813, 4.91469175444992974962636300619, 5.57893809623021291703244601007, 6.35325928031999033219085293354, 7.10144076013417515588056530153, 7.84854250279194552058818388017
