| L(s) = 1 | − 3-s − 7-s + 9-s − 2·13-s + 17-s + 2·19-s + 21-s − 6·23-s − 27-s − 6·29-s − 4·31-s + 4·37-s + 2·39-s + 6·41-s − 2·43-s + 49-s − 51-s − 2·57-s + 12·59-s + 8·61-s − 63-s − 14·67-s + 6·69-s − 2·73-s + 8·79-s + 81-s + 12·83-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.377·7-s + 1/3·9-s − 0.554·13-s + 0.242·17-s + 0.458·19-s + 0.218·21-s − 1.25·23-s − 0.192·27-s − 1.11·29-s − 0.718·31-s + 0.657·37-s + 0.320·39-s + 0.937·41-s − 0.304·43-s + 1/7·49-s − 0.140·51-s − 0.264·57-s + 1.56·59-s + 1.02·61-s − 0.125·63-s − 1.71·67-s + 0.722·69-s − 0.234·73-s + 0.900·79-s + 1/9·81-s + 1.31·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 17 | \( 1 - T \) | |
| good | 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 + 14 T + p T^{2} \) | 1.67.o |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−15.14052119859881, −14.62679711610428, −14.29021946638638, −13.37683471156656, −13.18104118352078, −12.48353741769818, −12.00325244667616, −11.59564307618421, −10.97099271581144, −10.42504278276694, −9.831492751448618, −9.478223786336379, −8.865949895978221, −8.025139721303073, −7.562163342710675, −7.062899332699531, −6.358919216468836, −5.777811829652884, −5.379762895483012, −4.622508688036923, −3.942910259589573, −3.420345825331447, −2.474924835870661, −1.863644076304356, −0.8535781618680735, 0,
0.8535781618680735, 1.863644076304356, 2.474924835870661, 3.420345825331447, 3.942910259589573, 4.622508688036923, 5.379762895483012, 5.777811829652884, 6.358919216468836, 7.062899332699531, 7.562163342710675, 8.025139721303073, 8.865949895978221, 9.478223786336379, 9.831492751448618, 10.42504278276694, 10.97099271581144, 11.59564307618421, 12.00325244667616, 12.48353741769818, 13.18104118352078, 13.37683471156656, 14.29021946638638, 14.62679711610428, 15.14052119859881
