| L(s) = 1 | − 4·5-s − 6·13-s − 8·17-s + 11·25-s + 4·29-s − 2·37-s + 8·41-s − 7·49-s + 4·53-s − 10·61-s + 24·65-s + 6·73-s + 32·85-s − 16·89-s − 18·97-s + 20·101-s − 6·109-s − 16·113-s + ⋯ |
| L(s) = 1 | − 1.78·5-s − 1.66·13-s − 1.94·17-s + 11/5·25-s + 0.742·29-s − 0.328·37-s + 1.24·41-s − 49-s + 0.549·53-s − 1.28·61-s + 2.97·65-s + 0.702·73-s + 3.47·85-s − 1.69·89-s − 1.82·97-s + 1.99·101-s − 0.574·109-s − 1.50·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| good | 5 | \( 1 + 4 T + p T^{2} \) | 1.5.e |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 + 8 T + p T^{2} \) | 1.17.i |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 8 T + p T^{2} \) | 1.41.ai |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 4 T + p T^{2} \) | 1.53.ae |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + p T^{2} \) | 1.67.a |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 16 T + p T^{2} \) | 1.89.q |
| 97 | \( 1 + 18 T + p T^{2} \) | 1.97.s |
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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
−11.43133431305815887793598810757, −10.64173856953551402265625207636, −9.327959901425306425198768202784, −8.338442505726918333995265218224, −7.46428835432640754113355987046, −6.69932041253260706274722644089, −4.85517883035181554193637871803, −4.15127486396668832984271170028, −2.68900975784905443020785166460, 0,
2.68900975784905443020785166460, 4.15127486396668832984271170028, 4.85517883035181554193637871803, 6.69932041253260706274722644089, 7.46428835432640754113355987046, 8.338442505726918333995265218224, 9.327959901425306425198768202784, 10.64173856953551402265625207636, 11.43133431305815887793598810757
