| L(s) = 1 | − 2·5-s − 4·7-s + 2·13-s + 2·17-s + 4·19-s + 23-s − 25-s − 2·29-s − 8·31-s + 8·35-s − 2·37-s − 10·41-s + 4·43-s + 9·49-s + 6·53-s + 12·59-s − 2·61-s − 4·65-s + 12·67-s + 16·71-s + 10·73-s − 4·79-s − 4·85-s − 6·89-s − 8·91-s − 8·95-s − 14·97-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 1.51·7-s + 0.554·13-s + 0.485·17-s + 0.917·19-s + 0.208·23-s − 1/5·25-s − 0.371·29-s − 1.43·31-s + 1.35·35-s − 0.328·37-s − 1.56·41-s + 0.609·43-s + 9/7·49-s + 0.824·53-s + 1.56·59-s − 0.256·61-s − 0.496·65-s + 1.46·67-s + 1.89·71-s + 1.17·73-s − 0.450·79-s − 0.433·85-s − 0.635·89-s − 0.838·91-s − 0.820·95-s − 1.42·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13248 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13248 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 23 | \( 1 - T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 - 16 T + p T^{2} \) | 1.71.aq |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.45122916811955, −15.91951751501409, −15.53977672803421, −14.99971366112324, −14.23430340761344, −13.60275376821073, −13.12985626886674, −12.46730266738866, −12.12142890651560, −11.35560109604673, −10.93010570100623, −10.05646816556074, −9.651255813272974, −9.060421509251111, −8.320470661649182, −7.746887596368898, −6.904793422985484, −6.751574779010674, −5.614800166217920, −5.360536155539788, −4.098316823409615, −3.573915619026354, −3.235755897158365, −2.163746205295502, −0.9307180587985533, 0,
0.9307180587985533, 2.163746205295502, 3.235755897158365, 3.573915619026354, 4.098316823409615, 5.360536155539788, 5.614800166217920, 6.751574779010674, 6.904793422985484, 7.746887596368898, 8.320470661649182, 9.060421509251111, 9.651255813272974, 10.05646816556074, 10.93010570100623, 11.35560109604673, 12.12142890651560, 12.46730266738866, 13.12985626886674, 13.60275376821073, 14.23430340761344, 14.99971366112324, 15.53977672803421, 15.91951751501409, 16.45122916811955
