| L(s) = 1 | − 3-s + 5-s + 9-s + 13-s − 15-s + 6·17-s + 4·23-s + 25-s − 27-s − 10·29-s − 6·37-s − 39-s − 2·41-s + 4·43-s + 45-s − 6·51-s − 6·53-s − 6·61-s + 65-s − 4·67-s − 4·69-s − 16·71-s + 2·73-s − 75-s + 81-s + 4·83-s + 6·85-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1/3·9-s + 0.277·13-s − 0.258·15-s + 1.45·17-s + 0.834·23-s + 1/5·25-s − 0.192·27-s − 1.85·29-s − 0.986·37-s − 0.160·39-s − 0.312·41-s + 0.609·43-s + 0.149·45-s − 0.840·51-s − 0.824·53-s − 0.768·61-s + 0.124·65-s − 0.488·67-s − 0.481·69-s − 1.89·71-s + 0.234·73-s − 0.115·75-s + 1/9·81-s + 0.439·83-s + 0.650·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 152880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 10 T + p T^{2} \) | 1.29.k |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 16 T + p T^{2} \) | 1.71.q |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−13.45499843453145, −13.05132997614353, −12.65016283764507, −12.01424210464165, −11.82550863087479, −10.94004142882082, −10.89263176719378, −10.23277389210344, −9.781365463675214, −9.235052820543349, −8.925879920481510, −8.160764623181117, −7.631134869934494, −7.233848962057689, −6.681875150613190, −6.062461630268379, −5.549268483758257, −5.366534911212166, −4.619957701617311, −4.057694274443592, −3.278249299377836, −3.049150747609050, −1.976086701149339, −1.548411735465971, −0.8699428032983438, 0,
0.8699428032983438, 1.548411735465971, 1.976086701149339, 3.049150747609050, 3.278249299377836, 4.057694274443592, 4.619957701617311, 5.366534911212166, 5.549268483758257, 6.062461630268379, 6.681875150613190, 7.233848962057689, 7.631134869934494, 8.160764623181117, 8.925879920481510, 9.235052820543349, 9.781365463675214, 10.23277389210344, 10.89263176719378, 10.94004142882082, 11.82550863087479, 12.01424210464165, 12.65016283764507, 13.05132997614353, 13.45499843453145
