| L(s) = 1 | − 3-s + 9-s − 11-s − 4·13-s + 2·17-s − 2·19-s − 2·23-s − 5·25-s − 27-s + 2·29-s − 8·31-s + 33-s + 6·37-s + 4·39-s − 6·41-s − 12·43-s + 6·47-s − 7·49-s − 2·51-s − 4·53-s + 2·57-s − 6·59-s + 4·61-s − 67-s + 2·69-s − 2·71-s − 10·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s − 0.301·11-s − 1.10·13-s + 0.485·17-s − 0.458·19-s − 0.417·23-s − 25-s − 0.192·27-s + 0.371·29-s − 1.43·31-s + 0.174·33-s + 0.986·37-s + 0.640·39-s − 0.937·41-s − 1.82·43-s + 0.875·47-s − 49-s − 0.280·51-s − 0.549·53-s + 0.264·57-s − 0.781·59-s + 0.512·61-s − 0.122·67-s + 0.240·69-s − 0.237·71-s − 1.17·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 141504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 141504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 67 | \( 1 + T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 + 4 T + p T^{2} \) | 1.53.e |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 - 4 T + p T^{2} \) | 1.61.ae |
| 71 | \( 1 + 2 T + p T^{2} \) | 1.71.c |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 8 T + p T^{2} \) | 1.83.ai |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−13.84688654570161, −13.31361787344970, −12.80275674743364, −12.49375123372675, −11.91443953982718, −11.50304704651366, −11.15407445656449, −10.37178552263628, −10.05686812085895, −9.762686104513821, −9.081629935596840, −8.533171245528032, −7.891211190975064, −7.527436402389414, −7.057995372692979, −6.399842459625210, −5.947731253858402, −5.425693445746679, −4.854311926430924, −4.501520164682343, −3.714696384269678, −3.238309073249737, −2.426569953274720, −1.890373278934409, −1.237321306857859, 0, 0,
1.237321306857859, 1.890373278934409, 2.426569953274720, 3.238309073249737, 3.714696384269678, 4.501520164682343, 4.854311926430924, 5.425693445746679, 5.947731253858402, 6.399842459625210, 7.057995372692979, 7.527436402389414, 7.891211190975064, 8.533171245528032, 9.081629935596840, 9.762686104513821, 10.05686812085895, 10.37178552263628, 11.15407445656449, 11.50304704651366, 11.91443953982718, 12.49375123372675, 12.80275674743364, 13.31361787344970, 13.84688654570161
