| L(s) = 1 | + 2·3-s + 5-s − 3·7-s + 9-s − 3·11-s + 2·15-s + 5·17-s − 19-s − 6·21-s − 4·25-s − 4·27-s + 2·29-s + 8·31-s − 6·33-s − 3·35-s + 10·37-s − 6·41-s + 7·43-s + 45-s − 9·47-s + 2·49-s + 10·51-s − 8·53-s − 3·55-s − 2·57-s + 14·59-s − 5·61-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.447·5-s − 1.13·7-s + 1/3·9-s − 0.904·11-s + 0.516·15-s + 1.21·17-s − 0.229·19-s − 1.30·21-s − 4/5·25-s − 0.769·27-s + 0.371·29-s + 1.43·31-s − 1.04·33-s − 0.507·35-s + 1.64·37-s − 0.937·41-s + 1.06·43-s + 0.149·45-s − 1.31·47-s + 2/7·49-s + 1.40·51-s − 1.09·53-s − 0.404·55-s − 0.264·57-s + 1.82·59-s − 0.640·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51376 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51376 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 13 | \( 1 \) | |
| 19 | \( 1 + T \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 17 | \( 1 - 5 T + p T^{2} \) | 1.17.af |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 7 T + p T^{2} \) | 1.43.ah |
| 47 | \( 1 + 9 T + p T^{2} \) | 1.47.j |
| 53 | \( 1 + 8 T + p T^{2} \) | 1.53.i |
| 59 | \( 1 - 14 T + p T^{2} \) | 1.59.ao |
| 61 | \( 1 + 5 T + p T^{2} \) | 1.61.f |
| 67 | \( 1 + p T^{2} \) | 1.67.a |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 - 15 T + p T^{2} \) | 1.73.ap |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 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
−14.77104604689850, −14.13593290198964, −13.65957265817058, −13.39769165524905, −12.79480356895990, −12.43842610723102, −11.71544772067783, −11.14264050045676, −10.29570064639683, −9.983107686511506, −9.554071552456853, −9.203113673418197, −8.273932626971139, −8.044916535329826, −7.628109679267157, −6.702171820419559, −6.313762747955432, −5.662369006900705, −5.128722343508129, −4.241824901122665, −3.631472828098535, −2.925398688862324, −2.738460443976280, −1.985645639009697, −1.026552671350696, 0,
1.026552671350696, 1.985645639009697, 2.738460443976280, 2.925398688862324, 3.631472828098535, 4.241824901122665, 5.128722343508129, 5.662369006900705, 6.313762747955432, 6.702171820419559, 7.628109679267157, 8.044916535329826, 8.273932626971139, 9.203113673418197, 9.554071552456853, 9.983107686511506, 10.29570064639683, 11.14264050045676, 11.71544772067783, 12.43842610723102, 12.79480356895990, 13.39769165524905, 13.65957265817058, 14.13593290198964, 14.77104604689850
