| L(s) = 1 | − 3-s − 5-s + 2·7-s + 9-s − 11-s + 13-s + 15-s − 4·17-s + 6·19-s − 2·21-s + 6·23-s + 25-s − 27-s − 8·29-s + 4·31-s + 33-s − 2·35-s + 2·37-s − 39-s − 6·41-s + 4·43-s − 45-s − 3·49-s + 4·51-s + 2·53-s + 55-s − 6·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.755·7-s + 1/3·9-s − 0.301·11-s + 0.277·13-s + 0.258·15-s − 0.970·17-s + 1.37·19-s − 0.436·21-s + 1.25·23-s + 1/5·25-s − 0.192·27-s − 1.48·29-s + 0.718·31-s + 0.174·33-s − 0.338·35-s + 0.328·37-s − 0.160·39-s − 0.937·41-s + 0.609·43-s − 0.149·45-s − 3/7·49-s + 0.560·51-s + 0.274·53-s + 0.134·55-s − 0.794·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 137280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 137280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 11 | \( 1 + T \) | |
| 13 | \( 1 - T \) | |
| good | 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 8 T + p T^{2} \) | 1.29.i |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + 16 T + p T^{2} \) | 1.71.q |
| 73 | \( 1 + 8 T + p T^{2} \) | 1.73.i |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 - 8 T + p T^{2} \) | 1.97.ai |
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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
−13.45099622722094, −13.23665451505324, −12.76514642027093, −12.11240216722121, −11.52998092149025, −11.33249806755735, −11.06844725965765, −10.29321714132760, −9.987482026400025, −9.157773970354903, −8.906098487743116, −8.269218550647894, −7.704123669428829, −7.280560771628767, −6.884194098775354, −6.209607320033081, −5.546775997097472, −5.213841911681162, −4.617131370419680, −4.215916784676618, −3.447389076754799, −2.939079618150641, −2.146452167568309, −1.436355337693911, −0.8513297524402985, 0,
0.8513297524402985, 1.436355337693911, 2.146452167568309, 2.939079618150641, 3.447389076754799, 4.215916784676618, 4.617131370419680, 5.213841911681162, 5.546775997097472, 6.209607320033081, 6.884194098775354, 7.280560771628767, 7.704123669428829, 8.269218550647894, 8.906098487743116, 9.157773970354903, 9.987482026400025, 10.29321714132760, 11.06844725965765, 11.33249806755735, 11.52998092149025, 12.11240216722121, 12.76514642027093, 13.23665451505324, 13.45099622722094
