| L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s + 3·7-s − 8-s + 9-s − 10-s + 3·11-s + 12-s − 5·13-s − 3·14-s + 15-s + 16-s + 8·17-s − 18-s − 19-s + 20-s + 3·21-s − 3·22-s − 23-s − 24-s − 4·25-s + 5·26-s + 27-s + 3·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 1.13·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.904·11-s + 0.288·12-s − 1.38·13-s − 0.801·14-s + 0.258·15-s + 1/4·16-s + 1.94·17-s − 0.235·18-s − 0.229·19-s + 0.223·20-s + 0.654·21-s − 0.639·22-s − 0.208·23-s − 0.204·24-s − 4/5·25-s + 0.980·26-s + 0.192·27-s + 0.566·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 255162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 255162 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 23 | \( 1 + T \) | |
| 43 | \( 1 \) | |
| good | 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 7 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 + 5 T + p T^{2} \) | 1.13.f |
| 17 | \( 1 - 8 T + p T^{2} \) | 1.17.ai |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 29 | \( 1 + T + p T^{2} \) | 1.29.b |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 47 | \( 1 - 5 T + p T^{2} \) | 1.47.af |
| 53 | \( 1 + 4 T + p T^{2} \) | 1.53.e |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 12 T + p T^{2} \) | 1.73.m |
| 79 | \( 1 + 16 T + p T^{2} \) | 1.79.q |
| 83 | \( 1 + T + p T^{2} \) | 1.83.b |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 7 T + p T^{2} \) | 1.97.ah |
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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
−13.06168696655476, −12.47423562973858, −11.98212544108127, −11.71873307971099, −11.32317867507868, −10.50468694171368, −10.18813779884936, −9.781600275279814, −9.376798188650664, −8.959449502858950, −8.297604702829249, −7.958291607986848, −7.557450950424034, −7.153219850660534, −6.597341524807885, −5.862510332096004, −5.436838700246812, −5.038825172739177, −4.103219391951257, −3.997518345885649, −3.008402161644326, −2.631877635420840, −1.907976971251771, −1.509816471253995, −1.011932714057804, 0,
1.011932714057804, 1.509816471253995, 1.907976971251771, 2.631877635420840, 3.008402161644326, 3.997518345885649, 4.103219391951257, 5.038825172739177, 5.436838700246812, 5.862510332096004, 6.597341524807885, 7.153219850660534, 7.557450950424034, 7.958291607986848, 8.297604702829249, 8.959449502858950, 9.376798188650664, 9.781600275279814, 10.18813779884936, 10.50468694171368, 11.32317867507868, 11.71873307971099, 11.98212544108127, 12.47423562973858, 13.06168696655476
