| L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s − 8-s + 9-s − 10-s − 12-s + 13-s − 15-s + 16-s − 2·17-s − 18-s + 8·19-s + 20-s − 4·23-s + 24-s + 25-s − 26-s − 27-s + 6·29-s + 30-s + 8·31-s − 32-s + 2·34-s + 36-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.288·12-s + 0.277·13-s − 0.258·15-s + 1/4·16-s − 0.485·17-s − 0.235·18-s + 1.83·19-s + 0.223·20-s − 0.834·23-s + 0.204·24-s + 1/5·25-s − 0.196·26-s − 0.192·27-s + 1.11·29-s + 0.182·30-s + 1.43·31-s − 0.176·32-s + 0.342·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.447735383\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.447735383\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 5 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
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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
−15.82694075493790, −15.46633885443823, −14.72372093978892, −13.81714526904698, −13.73838309033632, −12.98427185462881, −12.10148577268848, −11.80278013681367, −11.41242035067543, −10.51550536270879, −10.02375997128437, −9.870598637353296, −8.910018501766238, −8.483741699755445, −7.801632059895824, −7.085977822338760, −6.573778786498080, −6.032888261006813, −5.264985201045292, −4.813923424541594, −3.784097093108322, −3.053175781938190, −2.204526624120862, −1.358313610286355, −0.6276867804753838,
0.6276867804753838, 1.358313610286355, 2.204526624120862, 3.053175781938190, 3.784097093108322, 4.813923424541594, 5.264985201045292, 6.032888261006813, 6.573778786498080, 7.085977822338760, 7.801632059895824, 8.483741699755445, 8.910018501766238, 9.870598637353296, 10.02375997128437, 10.51550536270879, 11.41242035067543, 11.80278013681367, 12.10148577268848, 12.98427185462881, 13.73838309033632, 13.81714526904698, 14.72372093978892, 15.46633885443823, 15.82694075493790
