| L(s) = 1 | − 2·3-s + 2·5-s + 9-s + 4·11-s − 4·15-s + 6·17-s + 8·19-s − 23-s − 25-s + 4·27-s + 2·29-s + 6·31-s − 8·33-s + 2·37-s − 8·43-s + 2·45-s + 2·47-s − 12·51-s + 6·53-s + 8·55-s − 16·57-s − 6·59-s − 10·61-s − 12·67-s + 2·69-s + 16·71-s + 4·73-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.894·5-s + 1/3·9-s + 1.20·11-s − 1.03·15-s + 1.45·17-s + 1.83·19-s − 0.208·23-s − 1/5·25-s + 0.769·27-s + 0.371·29-s + 1.07·31-s − 1.39·33-s + 0.328·37-s − 1.21·43-s + 0.298·45-s + 0.291·47-s − 1.68·51-s + 0.824·53-s + 1.07·55-s − 2.11·57-s − 0.781·59-s − 1.28·61-s − 1.46·67-s + 0.240·69-s + 1.89·71-s + 0.468·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72128 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.655269365\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.655269365\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 23 | \( 1 + T \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 - 2 T + p T^{2} \) | 1.47.ac |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 - 16 T + p T^{2} \) | 1.71.aq |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−13.98624386626560, −13.76389230572532, −13.16830312199765, −12.22508889296009, −12.06834221722547, −11.80114060718006, −11.17648648806567, −10.58976004986034, −9.938381793386307, −9.762417930202422, −9.247457137641985, −8.524824513319773, −7.890725307613183, −7.295027043341109, −6.690236403038741, −6.150668739510183, −5.814092911319539, −5.301113524831161, −4.824964463371754, −4.081018370274736, −3.280445623987661, −2.851872393987968, −1.739597649083151, −1.204983390647701, −0.6663387939217298,
0.6663387939217298, 1.204983390647701, 1.739597649083151, 2.851872393987968, 3.280445623987661, 4.081018370274736, 4.824964463371754, 5.301113524831161, 5.814092911319539, 6.150668739510183, 6.690236403038741, 7.295027043341109, 7.890725307613183, 8.524824513319773, 9.247457137641985, 9.762417930202422, 9.938381793386307, 10.58976004986034, 11.17648648806567, 11.80114060718006, 12.06834221722547, 12.22508889296009, 13.16830312199765, 13.76389230572532, 13.98624386626560
