| L(s) = 1 | + 2·2-s − 3-s + 2·4-s − 2·6-s + 9-s + 6·11-s − 2·12-s + 2·13-s − 4·16-s + 2·17-s + 2·18-s − 4·19-s + 12·22-s + 6·23-s + 4·26-s − 27-s + 29-s − 5·31-s − 8·32-s − 6·33-s + 4·34-s + 2·36-s + 7·37-s − 8·38-s − 2·39-s + 10·41-s − 3·43-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 0.577·3-s + 4-s − 0.816·6-s + 1/3·9-s + 1.80·11-s − 0.577·12-s + 0.554·13-s − 16-s + 0.485·17-s + 0.471·18-s − 0.917·19-s + 2.55·22-s + 1.25·23-s + 0.784·26-s − 0.192·27-s + 0.185·29-s − 0.898·31-s − 1.41·32-s − 1.04·33-s + 0.685·34-s + 1/3·36-s + 1.15·37-s − 1.29·38-s − 0.320·39-s + 1.56·41-s − 0.457·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 106575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 106575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.658795383\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.658795383\) |
| \(L(\frac{3}{2})\) |
|
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 | 3 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 \) | |
| 29 | \( 1 - T \) | |
| good | 2 | \( 1 - p T + p T^{2} \) | 1.2.ac |
| 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 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.e |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 31 | \( 1 + 5 T + p T^{2} \) | 1.31.f |
| 37 | \( 1 - 7 T + p T^{2} \) | 1.37.ah |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 + 3 T + p T^{2} \) | 1.43.d |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 + 7 T + p T^{2} \) | 1.61.h |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + 4 T + p T^{2} \) | 1.71.e |
| 73 | \( 1 + 3 T + p T^{2} \) | 1.73.d |
| 79 | \( 1 - 11 T + p T^{2} \) | 1.79.al |
| 83 | \( 1 - 8 T + p T^{2} \) | 1.83.ai |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 97 | \( 1 + 15 T + p T^{2} \) | 1.97.p |
| 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
−13.70079892668144, −13.16441508446758, −12.63651147145832, −12.35232798658654, −11.95343086651111, −11.28583613074051, −10.94197779219717, −10.72442820416146, −9.521074906653844, −9.383610779135842, −8.888830626183629, −8.174848860131842, −7.409368950013555, −6.907260809636067, −6.335008931922267, −6.112949217067572, −5.579807657042044, −4.917186501975880, −4.284443793549238, −4.110811386445539, −3.415111851422813, −2.897334101804012, −2.027967551372944, −1.320326913347267, −0.6378512042497919,
0.6378512042497919, 1.320326913347267, 2.027967551372944, 2.897334101804012, 3.415111851422813, 4.110811386445539, 4.284443793549238, 4.917186501975880, 5.579807657042044, 6.112949217067572, 6.335008931922267, 6.907260809636067, 7.409368950013555, 8.174848860131842, 8.888830626183629, 9.383610779135842, 9.521074906653844, 10.72442820416146, 10.94197779219717, 11.28583613074051, 11.95343086651111, 12.35232798658654, 12.63651147145832, 13.16441508446758, 13.70079892668144
