| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s + 3·7-s − 8-s − 2·9-s + 12-s − 3·14-s + 16-s + 3·17-s + 2·18-s − 6·19-s + 3·21-s − 6·23-s − 24-s − 5·27-s + 3·28-s − 32-s − 3·34-s − 2·36-s + 3·37-s + 6·38-s − 3·42-s − 43-s + 6·46-s + 3·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s + 1.13·7-s − 0.353·8-s − 2/3·9-s + 0.288·12-s − 0.801·14-s + 1/4·16-s + 0.727·17-s + 0.471·18-s − 1.37·19-s + 0.654·21-s − 1.25·23-s − 0.204·24-s − 0.962·27-s + 0.566·28-s − 0.176·32-s − 0.514·34-s − 1/3·36-s + 0.493·37-s + 0.973·38-s − 0.462·42-s − 0.152·43-s + 0.884·46-s + 0.437·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.740245878\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.740245878\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 3 | \( 1 - T + p T^{2} \) | 1.3.ab |
| 7 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 3 T + p T^{2} \) | 1.37.ad |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 - 3 T + p T^{2} \) | 1.47.ad |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 - 15 T + p T^{2} \) | 1.71.ap |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 12 T + p T^{2} \) | 1.97.am |
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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
−8.167268080397728825770144395167, −7.40862748149144466958550408179, −6.46656684018924522714660519369, −5.81919936521029094824073749797, −5.06391040756493978960151315681, −4.14226598568978309926244161894, −3.40448287769266831131199619471, −2.29391370673890045834033086930, −1.94991966120951197998042855087, −0.68349243649354398924238112786,
0.68349243649354398924238112786, 1.94991966120951197998042855087, 2.29391370673890045834033086930, 3.40448287769266831131199619471, 4.14226598568978309926244161894, 5.06391040756493978960151315681, 5.81919936521029094824073749797, 6.46656684018924522714660519369, 7.40862748149144466958550408179, 8.167268080397728825770144395167
