| L(s) = 1 | − 3-s − 2·5-s − 4·7-s + 9-s − 11-s + 2·15-s + 6·17-s − 8·19-s + 4·21-s − 25-s − 27-s + 6·29-s + 33-s + 8·35-s + 6·37-s + 10·41-s + 8·43-s − 2·45-s + 9·49-s − 6·51-s − 6·53-s + 2·55-s + 8·57-s + 4·59-s + 2·61-s − 4·63-s − 12·67-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s − 1.51·7-s + 1/3·9-s − 0.301·11-s + 0.516·15-s + 1.45·17-s − 1.83·19-s + 0.872·21-s − 1/5·25-s − 0.192·27-s + 1.11·29-s + 0.174·33-s + 1.35·35-s + 0.986·37-s + 1.56·41-s + 1.21·43-s − 0.298·45-s + 9/7·49-s − 0.840·51-s − 0.824·53-s + 0.269·55-s + 1.05·57-s + 0.520·59-s + 0.256·61-s − 0.503·63-s − 1.46·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 356928 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 356928 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 11 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 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
−12.72444077450768, −12.34588170840394, −12.02020154388493, −11.31990020850676, −11.03612406162401, −10.42885964515850, −10.11229641023157, −9.717906767561937, −9.144957172721737, −8.699568630605942, −7.953341816110386, −7.819643472684947, −7.182071830818957, −6.714486004420447, −6.138557996152687, −5.961252636419485, −5.407842717601249, −4.556611183463631, −4.238058907678181, −3.824267926399901, −3.140732411254940, −2.767214322844468, −2.135811968065227, −1.135739524438072, −0.5890164102584634, 0,
0.5890164102584634, 1.135739524438072, 2.135811968065227, 2.767214322844468, 3.140732411254940, 3.824267926399901, 4.238058907678181, 4.556611183463631, 5.407842717601249, 5.961252636419485, 6.138557996152687, 6.714486004420447, 7.182071830818957, 7.819643472684947, 7.953341816110386, 8.699568630605942, 9.144957172721737, 9.717906767561937, 10.11229641023157, 10.42885964515850, 11.03612406162401, 11.31990020850676, 12.02020154388493, 12.34588170840394, 12.72444077450768
