| L(s) = 1 | + 3-s + 5-s + 7-s + 9-s − 5·13-s + 15-s + 6·17-s + 19-s + 21-s − 4·23-s − 4·25-s + 27-s + 29-s + 10·31-s + 35-s − 37-s − 5·39-s + 8·43-s + 45-s − 47-s + 49-s + 6·51-s − 8·53-s + 57-s − 3·59-s − 6·61-s + 63-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 0.377·7-s + 1/3·9-s − 1.38·13-s + 0.258·15-s + 1.45·17-s + 0.229·19-s + 0.218·21-s − 0.834·23-s − 4/5·25-s + 0.192·27-s + 0.185·29-s + 1.79·31-s + 0.169·35-s − 0.164·37-s − 0.800·39-s + 1.21·43-s + 0.149·45-s − 0.145·47-s + 1/7·49-s + 0.840·51-s − 1.09·53-s + 0.132·57-s − 0.390·59-s − 0.768·61-s + 0.125·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 11 | \( 1 \) | |
| good | 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 13 | \( 1 + 5 T + p T^{2} \) | 1.13.f |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - T + p T^{2} \) | 1.29.ab |
| 31 | \( 1 - 10 T + p T^{2} \) | 1.31.ak |
| 37 | \( 1 + T + p T^{2} \) | 1.37.b |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + T + p T^{2} \) | 1.47.b |
| 53 | \( 1 + 8 T + p T^{2} \) | 1.53.i |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 - 13 T + p T^{2} \) | 1.67.an |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 3 T + p T^{2} \) | 1.73.ad |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 2 T + p T^{2} \) | 1.83.c |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 16 T + p T^{2} \) | 1.97.q |
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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
−13.83394826150760, −12.97097818406574, −12.52477992607561, −12.06887451324884, −11.82432153386919, −11.12289810863990, −10.47643177846714, −9.998540402395794, −9.662238156249626, −9.465091111564281, −8.563887561892458, −8.175233961977163, −7.645721367717125, −7.453085255716268, −6.681613630811341, −6.103382516609806, −5.626022144407957, −5.029145973970761, −4.575580206297715, −3.975404845589896, −3.330478881038387, −2.665388317149781, −2.340245404620454, −1.563265391923295, −1.003727787824515, 0,
1.003727787824515, 1.563265391923295, 2.340245404620454, 2.665388317149781, 3.330478881038387, 3.975404845589896, 4.575580206297715, 5.029145973970761, 5.626022144407957, 6.103382516609806, 6.681613630811341, 7.453085255716268, 7.645721367717125, 8.175233961977163, 8.563887561892458, 9.465091111564281, 9.662238156249626, 9.998540402395794, 10.47643177846714, 11.12289810863990, 11.82432153386919, 12.06887451324884, 12.52477992607561, 12.97097818406574, 13.83394826150760
