| L(s) = 1 | − 3-s − 5-s + 9-s + 2·11-s − 13-s + 15-s + 4·17-s − 19-s − 4·23-s + 25-s − 27-s − 5·31-s − 2·33-s − 5·37-s + 39-s − 2·41-s + 9·43-s − 45-s − 2·47-s − 4·51-s + 12·53-s − 2·55-s + 57-s − 8·59-s + 14·61-s + 65-s − 9·67-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1/3·9-s + 0.603·11-s − 0.277·13-s + 0.258·15-s + 0.970·17-s − 0.229·19-s − 0.834·23-s + 1/5·25-s − 0.192·27-s − 0.898·31-s − 0.348·33-s − 0.821·37-s + 0.160·39-s − 0.312·41-s + 1.37·43-s − 0.149·45-s − 0.291·47-s − 0.560·51-s + 1.64·53-s − 0.269·55-s + 0.132·57-s − 1.04·59-s + 1.79·61-s + 0.124·65-s − 1.09·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| good | 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 5 T + p T^{2} \) | 1.31.f |
| 37 | \( 1 + 5 T + p T^{2} \) | 1.37.f |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 9 T + p T^{2} \) | 1.43.aj |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 + 9 T + p T^{2} \) | 1.67.j |
| 71 | \( 1 + 2 T + p T^{2} \) | 1.71.c |
| 73 | \( 1 + T + p T^{2} \) | 1.73.b |
| 79 | \( 1 - 3 T + p T^{2} \) | 1.79.ad |
| 83 | \( 1 + 18 T + p T^{2} \) | 1.83.s |
| 89 | \( 1 - 4 T + p T^{2} \) | 1.89.ae |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−16.74278396029709, −16.15165207876881, −15.68678154639689, −14.95516514117581, −14.46337922282086, −13.96447524096665, −13.16246688203386, −12.53740699781520, −12.02465784096418, −11.68801387079127, −10.93054287276210, −10.37839360699780, −9.806077893289631, −9.116511087123677, −8.458192719083664, −7.728905159010457, −7.174238285044784, −6.581206398643339, −5.710116464853148, −5.364075029298547, −4.324454038158305, −3.905593464683194, −3.051688816737260, −1.995322213233195, −1.083472587763492, 0,
1.083472587763492, 1.995322213233195, 3.051688816737260, 3.905593464683194, 4.324454038158305, 5.364075029298547, 5.710116464853148, 6.581206398643339, 7.174238285044784, 7.728905159010457, 8.458192719083664, 9.116511087123677, 9.806077893289631, 10.37839360699780, 10.93054287276210, 11.68801387079127, 12.02465784096418, 12.53740699781520, 13.16246688203386, 13.96447524096665, 14.46337922282086, 14.95516514117581, 15.68678154639689, 16.15165207876881, 16.74278396029709
