| L(s) = 1 | − 2-s + 4-s − 5·7-s − 8-s − 2·11-s + 5·14-s + 16-s − 2·17-s + 3·19-s + 2·22-s + 6·23-s − 5·28-s − 10·29-s − 32-s + 2·34-s + 2·37-s − 3·38-s − 8·41-s + 43-s − 2·44-s − 6·46-s + 12·47-s + 18·49-s − 2·53-s + 5·56-s + 10·58-s − 14·59-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 1.88·7-s − 0.353·8-s − 0.603·11-s + 1.33·14-s + 1/4·16-s − 0.485·17-s + 0.688·19-s + 0.426·22-s + 1.25·23-s − 0.944·28-s − 1.85·29-s − 0.176·32-s + 0.342·34-s + 0.328·37-s − 0.486·38-s − 1.24·41-s + 0.152·43-s − 0.301·44-s − 0.884·46-s + 1.75·47-s + 18/7·49-s − 0.274·53-s + 0.668·56-s + 1.31·58-s − 1.82·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 3 | \( 1 \) | |
| 5 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 7 | \( 1 + 5 T + p T^{2} \) | 1.7.f |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 3 T + p T^{2} \) | 1.19.ad |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 10 T + p T^{2} \) | 1.29.k |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 8 T + p T^{2} \) | 1.41.i |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + 14 T + p T^{2} \) | 1.59.o |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 9 T + p T^{2} \) | 1.67.aj |
| 71 | \( 1 - 4 T + p T^{2} \) | 1.71.ae |
| 73 | \( 1 + T + p T^{2} \) | 1.73.b |
| 79 | \( 1 - 11 T + p T^{2} \) | 1.79.al |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−14.23529597100580, −13.64198891971894, −13.23647610619728, −12.76850222298866, −12.44245884133458, −11.76121469782106, −11.16591911679539, −10.66758222757200, −10.28704366205290, −9.515197439648512, −9.400591581066139, −8.946246207064823, −8.259247203265410, −7.437223933251606, −7.290624454655698, −6.586328494992687, −6.212628295807959, −5.481575207326554, −5.104206610332828, −4.000829344188411, −3.567778556477771, −2.869408738135376, −2.532200044283509, −1.583492682338162, −0.6548165172368694, 0,
0.6548165172368694, 1.583492682338162, 2.532200044283509, 2.869408738135376, 3.567778556477771, 4.000829344188411, 5.104206610332828, 5.481575207326554, 6.212628295807959, 6.586328494992687, 7.290624454655698, 7.437223933251606, 8.259247203265410, 8.946246207064823, 9.400591581066139, 9.515197439648512, 10.28704366205290, 10.66758222757200, 11.16591911679539, 11.76121469782106, 12.44245884133458, 12.76850222298866, 13.23647610619728, 13.64198891971894, 14.23529597100580
