| L(s) = 1 | + 3-s − 5-s + 7-s + 9-s + 13-s − 15-s − 4·17-s + 21-s + 7·23-s − 4·25-s + 27-s − 8·29-s + 10·31-s − 35-s − 8·37-s + 39-s + 6·43-s − 45-s + 8·47-s + 49-s − 4·51-s − 10·53-s + 5·59-s − 7·61-s + 63-s − 65-s − 14·67-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s + 0.277·13-s − 0.258·15-s − 0.970·17-s + 0.218·21-s + 1.45·23-s − 4/5·25-s + 0.192·27-s − 1.48·29-s + 1.79·31-s − 0.169·35-s − 1.31·37-s + 0.160·39-s + 0.914·43-s − 0.149·45-s + 1.16·47-s + 1/7·49-s − 0.560·51-s − 1.37·53-s + 0.650·59-s − 0.896·61-s + 0.125·63-s − 0.124·65-s − 1.71·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 121296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121296 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 19 | \( 1 \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 23 | \( 1 - 7 T + p T^{2} \) | 1.23.ah |
| 29 | \( 1 + 8 T + p T^{2} \) | 1.29.i |
| 31 | \( 1 - 10 T + p T^{2} \) | 1.31.ak |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 - 5 T + p T^{2} \) | 1.59.af |
| 61 | \( 1 + 7 T + p T^{2} \) | 1.61.h |
| 67 | \( 1 + 14 T + p T^{2} \) | 1.67.o |
| 71 | \( 1 + 7 T + p T^{2} \) | 1.71.h |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 3 T + p T^{2} \) | 1.83.ad |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 18 T + p T^{2} \) | 1.97.as |
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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
−13.76031533974490, −13.40677196421451, −12.92488540147830, −12.32634261088497, −11.90658014369669, −11.28050595274925, −10.97384855701015, −10.47432970825307, −9.867107019322111, −9.234129986701828, −8.855032129431765, −8.524464601191198, −7.802310220354752, −7.438300731571029, −7.027525975307251, −6.269234340057714, −5.877835876113910, −4.996334664566834, −4.654927945848622, −4.044719401387941, −3.510762860859998, −2.913934578467188, −2.273076799879077, −1.655145188701694, −0.9205699640415479, 0,
0.9205699640415479, 1.655145188701694, 2.273076799879077, 2.913934578467188, 3.510762860859998, 4.044719401387941, 4.654927945848622, 4.996334664566834, 5.877835876113910, 6.269234340057714, 7.027525975307251, 7.438300731571029, 7.802310220354752, 8.524464601191198, 8.855032129431765, 9.234129986701828, 9.867107019322111, 10.47432970825307, 10.97384855701015, 11.28050595274925, 11.90658014369669, 12.32634261088497, 12.92488540147830, 13.40677196421451, 13.76031533974490
