| L(s) = 1 | + 2-s + 4-s + 7-s + 8-s − 3·9-s + 4·11-s − 6·13-s + 14-s + 16-s − 2·17-s − 3·18-s + 4·22-s − 6·26-s + 28-s − 6·29-s − 8·31-s + 32-s − 2·34-s − 3·36-s − 10·37-s − 2·41-s − 4·43-s + 4·44-s − 8·47-s + 49-s − 6·52-s − 2·53-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.377·7-s + 0.353·8-s − 9-s + 1.20·11-s − 1.66·13-s + 0.267·14-s + 1/4·16-s − 0.485·17-s − 0.707·18-s + 0.852·22-s − 1.17·26-s + 0.188·28-s − 1.11·29-s − 1.43·31-s + 0.176·32-s − 0.342·34-s − 1/2·36-s − 1.64·37-s − 0.312·41-s − 0.609·43-s + 0.603·44-s − 1.16·47-s + 1/7·49-s − 0.832·52-s − 0.274·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 126350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.286470738\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.286470738\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| 19 | \( 1 \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 - 16 T + p T^{2} \) | 1.71.aq |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 8 T + p T^{2} \) | 1.83.i |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 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
−13.67370841310706, −12.95617464939165, −12.55094780966698, −11.95916067868648, −11.78406752308957, −11.16123729038562, −10.88726969859489, −10.12410115398561, −9.636394446124105, −9.008018086415247, −8.792704268184567, −7.974254397306508, −7.524200035500327, −6.927580372182020, −6.623645625332017, −5.876394397229312, −5.390033833532099, −4.946315321266898, −4.463951954690294, −3.664113630058706, −3.366881596218166, −2.595344481448699, −1.922629195862802, −1.577192982627870, −0.2765442847958913,
0.2765442847958913, 1.577192982627870, 1.922629195862802, 2.595344481448699, 3.366881596218166, 3.664113630058706, 4.463951954690294, 4.946315321266898, 5.390033833532099, 5.876394397229312, 6.623645625332017, 6.927580372182020, 7.524200035500327, 7.974254397306508, 8.792704268184567, 9.008018086415247, 9.636394446124105, 10.12410115398561, 10.88726969859489, 11.16123729038562, 11.78406752308957, 11.95916067868648, 12.55094780966698, 12.95617464939165, 13.67370841310706
