| L(s) = 1 | − 3-s − 2·7-s + 9-s − 11-s − 4·13-s − 4·17-s + 2·21-s − 4·23-s − 27-s + 6·29-s + 8·31-s + 33-s − 2·37-s + 4·39-s − 10·41-s − 10·43-s − 12·47-s − 3·49-s + 4·51-s + 6·53-s − 12·59-s − 10·61-s − 2·63-s + 8·67-s + 4·69-s − 4·73-s + 2·77-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.755·7-s + 1/3·9-s − 0.301·11-s − 1.10·13-s − 0.970·17-s + 0.436·21-s − 0.834·23-s − 0.192·27-s + 1.11·29-s + 1.43·31-s + 0.174·33-s − 0.328·37-s + 0.640·39-s − 1.56·41-s − 1.52·43-s − 1.75·47-s − 3/7·49-s + 0.560·51-s + 0.824·53-s − 1.56·59-s − 1.28·61-s − 0.251·63-s + 0.977·67-s + 0.481·69-s − 0.468·73-s + 0.227·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 52800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 52800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + 2 T + p T^{2} \) | 1.83.c |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 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
−15.17434915253380, −14.42865450508857, −13.81942832315170, −13.43330893676264, −12.89323592845065, −12.38155556739558, −11.80318491120325, −11.64825028002908, −10.78300900386080, −10.17860257737033, −9.976885704927926, −9.482940057324331, −8.629088416600439, −8.231719794911262, −7.590934239158566, −6.777542476993487, −6.591518588257089, −6.079974293936501, −5.180104056209355, −4.797083336890517, −4.311207619222351, −3.358897938335645, −2.873365351132614, −2.112784108987635, −1.345501505185684, 0, 0,
1.345501505185684, 2.112784108987635, 2.873365351132614, 3.358897938335645, 4.311207619222351, 4.797083336890517, 5.180104056209355, 6.079974293936501, 6.591518588257089, 6.777542476993487, 7.590934239158566, 8.231719794911262, 8.629088416600439, 9.482940057324331, 9.976885704927926, 10.17860257737033, 10.78300900386080, 11.64825028002908, 11.80318491120325, 12.38155556739558, 12.89323592845065, 13.43330893676264, 13.81942832315170, 14.42865450508857, 15.17434915253380
