| L(s) = 1 | − 3-s − 2·7-s + 9-s − 3·11-s − 13-s − 3·17-s − 5·19-s + 2·21-s + 4·23-s − 27-s − 31-s + 3·33-s − 2·37-s + 39-s + 2·41-s + 6·43-s − 7·47-s − 3·49-s + 3·51-s + 14·53-s + 5·57-s + 10·59-s − 7·61-s − 2·63-s + 7·67-s − 4·69-s + 3·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.755·7-s + 1/3·9-s − 0.904·11-s − 0.277·13-s − 0.727·17-s − 1.14·19-s + 0.436·21-s + 0.834·23-s − 0.192·27-s − 0.179·31-s + 0.522·33-s − 0.328·37-s + 0.160·39-s + 0.312·41-s + 0.914·43-s − 1.02·47-s − 3/7·49-s + 0.420·51-s + 1.92·53-s + 0.662·57-s + 1.30·59-s − 0.896·61-s − 0.251·63-s + 0.855·67-s − 0.481·69-s + 0.356·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 148800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 148800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5651283941\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5651283941\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 31 | \( 1 + T \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 + 5 T + p T^{2} \) | 1.19.f |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 + 7 T + p T^{2} \) | 1.47.h |
| 53 | \( 1 - 14 T + p T^{2} \) | 1.53.ao |
| 59 | \( 1 - 10 T + p T^{2} \) | 1.59.ak |
| 61 | \( 1 + 7 T + p T^{2} \) | 1.61.h |
| 67 | \( 1 - 7 T + p T^{2} \) | 1.67.ah |
| 71 | \( 1 - 3 T + p T^{2} \) | 1.71.ad |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + 15 T + p T^{2} \) | 1.79.p |
| 83 | \( 1 - T + p T^{2} \) | 1.83.ab |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 + 13 T + p T^{2} \) | 1.97.n |
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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
−13.23369945388639, −12.81357016198546, −12.60704126255012, −11.90106225053924, −11.41156303764220, −10.90299325380502, −10.51398001436580, −10.12076028661233, −9.546086999813702, −9.036978964059791, −8.549181258393165, −7.997203300122472, −7.383681499908426, −6.827532752090227, −6.567192052453934, −5.921044346220801, −5.385277964390915, −4.943554285315879, −4.302955801762009, −3.828500014645024, −3.075652184575649, −2.477681324422481, −2.024594162686977, −1.032079503267415, −0.2560825865981520,
0.2560825865981520, 1.032079503267415, 2.024594162686977, 2.477681324422481, 3.075652184575649, 3.828500014645024, 4.302955801762009, 4.943554285315879, 5.385277964390915, 5.921044346220801, 6.567192052453934, 6.827532752090227, 7.383681499908426, 7.997203300122472, 8.549181258393165, 9.036978964059791, 9.546086999813702, 10.12076028661233, 10.51398001436580, 10.90299325380502, 11.41156303764220, 11.90106225053924, 12.60704126255012, 12.81357016198546, 13.23369945388639
