| L(s) = 1 | − 4·5-s + 2·11-s − 6·13-s − 4·17-s − 4·19-s − 2·23-s + 11·25-s − 2·29-s − 2·37-s + 4·43-s + 12·47-s − 6·53-s − 8·55-s + 8·59-s + 6·61-s + 24·65-s + 8·67-s − 14·71-s + 2·73-s + 12·79-s + 4·83-s + 16·85-s + 16·95-s + 2·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 1.78·5-s + 0.603·11-s − 1.66·13-s − 0.970·17-s − 0.917·19-s − 0.417·23-s + 11/5·25-s − 0.371·29-s − 0.328·37-s + 0.609·43-s + 1.75·47-s − 0.824·53-s − 1.07·55-s + 1.04·59-s + 0.768·61-s + 2.97·65-s + 0.977·67-s − 1.66·71-s + 0.234·73-s + 1.35·79-s + 0.439·83-s + 1.73·85-s + 1.64·95-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| good | 5 | \( 1 + 4 T + p T^{2} \) | 1.5.e |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + 14 T + p T^{2} \) | 1.71.o |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 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
−15.48792078216799, −14.95906308459183, −14.58998101490577, −14.08731061941481, −13.21745575258974, −12.59476752208547, −12.26145851693873, −11.77568093124447, −11.28920073050997, −10.74848168521445, −10.18466151643324, −9.373582675945452, −8.890660731679751, −8.306708799658545, −7.748005788590851, −7.173085750386513, −6.864124301618197, −6.067518788249061, −5.126757421327084, −4.536157782334019, −4.097819038680767, −3.568574691058031, −2.627217988459339, −2.048009564992990, −0.7134336306824323, 0,
0.7134336306824323, 2.048009564992990, 2.627217988459339, 3.568574691058031, 4.097819038680767, 4.536157782334019, 5.126757421327084, 6.067518788249061, 6.864124301618197, 7.173085750386513, 7.748005788590851, 8.306708799658545, 8.890660731679751, 9.373582675945452, 10.18466151643324, 10.74848168521445, 11.28920073050997, 11.77568093124447, 12.26145851693873, 12.59476752208547, 13.21745575258974, 14.08731061941481, 14.58998101490577, 14.95906308459183, 15.48792078216799
