| L(s) = 1 | − 3-s + 2·5-s + 9-s + 11-s − 2·13-s − 2·15-s − 2·17-s + 4·19-s + 4·23-s − 25-s − 27-s − 6·29-s + 4·31-s − 33-s − 2·37-s + 2·39-s + 6·41-s + 12·43-s + 2·45-s + 4·47-s + 2·51-s + 6·53-s + 2·55-s − 4·57-s − 4·59-s − 2·61-s − 4·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s + 1/3·9-s + 0.301·11-s − 0.554·13-s − 0.516·15-s − 0.485·17-s + 0.917·19-s + 0.834·23-s − 1/5·25-s − 0.192·27-s − 1.11·29-s + 0.718·31-s − 0.174·33-s − 0.328·37-s + 0.320·39-s + 0.937·41-s + 1.82·43-s + 0.298·45-s + 0.583·47-s + 0.280·51-s + 0.824·53-s + 0.269·55-s − 0.529·57-s − 0.520·59-s − 0.256·61-s − 0.496·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 18 T + p T^{2} \) | 1.97.s |
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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
−14.67460483048176, −14.15263395344230, −13.73591603417303, −13.11416058851924, −12.84508032787642, −11.98888150293198, −11.83956728470555, −11.04752901850393, −10.67858676970780, −10.09337168806530, −9.504281299751319, −9.191674995910255, −8.672841244654945, −7.616285772088970, −7.406111161727546, −6.761413003751651, −6.065261701996373, −5.661150946106332, −5.237242493920405, −4.402893173155771, −4.033137733629300, −2.950817969071978, −2.514699421755093, −1.624993015999444, −1.046549400889553, 0,
1.046549400889553, 1.624993015999444, 2.514699421755093, 2.950817969071978, 4.033137733629300, 4.402893173155771, 5.237242493920405, 5.661150946106332, 6.065261701996373, 6.761413003751651, 7.406111161727546, 7.616285772088970, 8.672841244654945, 9.191674995910255, 9.504281299751319, 10.09337168806530, 10.67858676970780, 11.04752901850393, 11.83956728470555, 11.98888150293198, 12.84508032787642, 13.11416058851924, 13.73591603417303, 14.15263395344230, 14.67460483048176
