| L(s) = 1 | − 2·3-s + 3·7-s + 9-s − 3·11-s + 13-s + 2·17-s + 19-s − 6·21-s − 23-s + 4·27-s + 9·29-s + 10·31-s + 6·33-s + 4·37-s − 2·39-s − 41-s − 9·43-s + 6·47-s + 2·49-s − 4·51-s − 8·53-s − 2·57-s − 8·59-s − 8·61-s + 3·63-s − 4·67-s + 2·69-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1.13·7-s + 1/3·9-s − 0.904·11-s + 0.277·13-s + 0.485·17-s + 0.229·19-s − 1.30·21-s − 0.208·23-s + 0.769·27-s + 1.67·29-s + 1.79·31-s + 1.04·33-s + 0.657·37-s − 0.320·39-s − 0.156·41-s − 1.37·43-s + 0.875·47-s + 2/7·49-s − 0.560·51-s − 1.09·53-s − 0.264·57-s − 1.04·59-s − 1.02·61-s + 0.377·63-s − 0.488·67-s + 0.240·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 23 | \( 1 + T \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 7 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 29 | \( 1 - 9 T + p T^{2} \) | 1.29.aj |
| 31 | \( 1 - 10 T + p T^{2} \) | 1.31.ak |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 + T + p T^{2} \) | 1.41.b |
| 43 | \( 1 + 9 T + p T^{2} \) | 1.43.j |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 + 8 T + p T^{2} \) | 1.53.i |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 9 T + p T^{2} \) | 1.73.j |
| 79 | \( 1 + 3 T + p T^{2} \) | 1.79.d |
| 83 | \( 1 + 5 T + p T^{2} \) | 1.83.f |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−15.33714932365992, −14.52631673321247, −14.13478711981341, −13.58637974993013, −13.05977321220284, −12.18555580900016, −12.01376711692795, −11.53700674203965, −10.94027481467040, −10.44976023600140, −10.16687908261328, −9.372493108335256, −8.494970348332604, −8.133204043632144, −7.724100609941148, −6.870488400619947, −6.292079655254633, −5.809162238108665, −5.153337501219327, −4.710387544177149, −4.330223284407311, −3.087820848530894, −2.677796434401340, −1.538278758051497, −0.9944825845128874, 0,
0.9944825845128874, 1.538278758051497, 2.677796434401340, 3.087820848530894, 4.330223284407311, 4.710387544177149, 5.153337501219327, 5.809162238108665, 6.292079655254633, 6.870488400619947, 7.724100609941148, 8.133204043632144, 8.494970348332604, 9.372493108335256, 10.16687908261328, 10.44976023600140, 10.94027481467040, 11.53700674203965, 12.01376711692795, 12.18555580900016, 13.05977321220284, 13.58637974993013, 14.13478711981341, 14.52631673321247, 15.33714932365992
