| L(s) = 1 | − 3-s + 2·7-s + 9-s − 11-s + 2·13-s − 17-s − 4·19-s − 2·21-s + 6·23-s − 5·25-s − 27-s − 5·29-s + 3·31-s + 33-s + 3·37-s − 2·39-s + 41-s − 7·43-s + 3·47-s − 3·49-s + 51-s − 10·53-s + 4·57-s − 61-s + 2·63-s + 2·67-s − 6·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.755·7-s + 1/3·9-s − 0.301·11-s + 0.554·13-s − 0.242·17-s − 0.917·19-s − 0.436·21-s + 1.25·23-s − 25-s − 0.192·27-s − 0.928·29-s + 0.538·31-s + 0.174·33-s + 0.493·37-s − 0.320·39-s + 0.156·41-s − 1.06·43-s + 0.437·47-s − 3/7·49-s + 0.140·51-s − 1.37·53-s + 0.529·57-s − 0.128·61-s + 0.251·63-s + 0.244·67-s − 0.722·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7872 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 41 | \( 1 - T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + T + p T^{2} \) | 1.11.b |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + T + p T^{2} \) | 1.17.b |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 5 T + p T^{2} \) | 1.29.f |
| 31 | \( 1 - 3 T + p T^{2} \) | 1.31.ad |
| 37 | \( 1 - 3 T + p T^{2} \) | 1.37.ad |
| 43 | \( 1 + 7 T + p T^{2} \) | 1.43.h |
| 47 | \( 1 - 3 T + p T^{2} \) | 1.47.ad |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + T + p T^{2} \) | 1.61.b |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 + 3 T + p T^{2} \) | 1.71.d |
| 73 | \( 1 + 11 T + p T^{2} \) | 1.73.l |
| 79 | \( 1 + 6 T + p T^{2} \) | 1.79.g |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 8 T + p T^{2} \) | 1.97.ai |
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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
−7.56636400347068451470053269646, −6.71193474496447008855501085183, −6.12052656454524586273868181391, −5.38985998961051428608330715209, −4.71618595815485575158607642279, −4.09723514241314850048289335463, −3.13432764848909093248569194024, −2.07018754704280249105513554095, −1.27950630160954544082498454934, 0,
1.27950630160954544082498454934, 2.07018754704280249105513554095, 3.13432764848909093248569194024, 4.09723514241314850048289335463, 4.71618595815485575158607642279, 5.38985998961051428608330715209, 6.12052656454524586273868181391, 6.71193474496447008855501085183, 7.56636400347068451470053269646
