| L(s) = 1 | − 3-s − 5-s + 9-s − 11-s − 2·13-s + 15-s + 4·19-s + 8·23-s + 25-s − 27-s − 6·29-s + 33-s + 10·37-s + 2·39-s + 6·41-s − 4·43-s − 45-s − 7·49-s + 6·53-s + 55-s − 4·57-s + 4·59-s − 14·61-s + 2·65-s + 4·67-s − 8·69-s − 10·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1/3·9-s − 0.301·11-s − 0.554·13-s + 0.258·15-s + 0.917·19-s + 1.66·23-s + 1/5·25-s − 0.192·27-s − 1.11·29-s + 0.174·33-s + 1.64·37-s + 0.320·39-s + 0.937·41-s − 0.609·43-s − 0.149·45-s − 49-s + 0.824·53-s + 0.134·55-s − 0.529·57-s + 0.520·59-s − 1.79·61-s + 0.248·65-s + 0.488·67-s − 0.963·69-s − 1.17·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 381480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 381480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.466879369\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.466879369\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 11 | \( 1 + T \) | |
| 17 | \( 1 \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
| 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
−12.52358195744618, −11.82683230228553, −11.65075552516784, −11.12633276366945, −10.81340548644958, −10.29820482108668, −9.719627430411540, −9.324155949838761, −9.010682951340260, −8.289671345508096, −7.729205197710159, −7.443344457581140, −7.081132471513745, −6.432614284372514, −5.988678579444048, −5.399487133071936, −4.993054447687773, −4.618280612475052, −3.998093652908522, −3.428323121691605, −2.870328559439989, −2.423062105743761, −1.553617398756344, −0.9994456171340769, −0.3765272166989982,
0.3765272166989982, 0.9994456171340769, 1.553617398756344, 2.423062105743761, 2.870328559439989, 3.428323121691605, 3.998093652908522, 4.618280612475052, 4.993054447687773, 5.399487133071936, 5.988678579444048, 6.432614284372514, 7.081132471513745, 7.443344457581140, 7.729205197710159, 8.289671345508096, 9.010682951340260, 9.324155949838761, 9.719627430411540, 10.29820482108668, 10.81340548644958, 11.12633276366945, 11.65075552516784, 11.82683230228553, 12.52358195744618
