| L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s − 4·7-s − 8-s + 9-s − 10-s + 5·11-s − 12-s − 13-s + 4·14-s − 15-s + 16-s − 18-s − 19-s + 20-s + 4·21-s − 5·22-s + 2·23-s + 24-s + 25-s + 26-s − 27-s − 4·28-s + 8·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 1.51·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 1.50·11-s − 0.288·12-s − 0.277·13-s + 1.06·14-s − 0.258·15-s + 1/4·16-s − 0.235·18-s − 0.229·19-s + 0.223·20-s + 0.872·21-s − 1.06·22-s + 0.417·23-s + 0.204·24-s + 1/5·25-s + 0.196·26-s − 0.192·27-s − 0.755·28-s + 1.48·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6317516090\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6317516090\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 13 | \( 1 + T \) | |
| 17 | \( 1 \) | |
| good | 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 - 5 T + p T^{2} \) | 1.11.af |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 - 8 T + p T^{2} \) | 1.29.ai |
| 31 | \( 1 + 7 T + p T^{2} \) | 1.31.h |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 13 T + p T^{2} \) | 1.43.n |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 5 T + p T^{2} \) | 1.61.f |
| 67 | \( 1 + 14 T + p T^{2} \) | 1.67.o |
| 71 | \( 1 + 14 T + p T^{2} \) | 1.71.o |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + 2 T + p T^{2} \) | 1.79.c |
| 83 | \( 1 + 9 T + p T^{2} \) | 1.83.j |
| 89 | \( 1 + 9 T + p T^{2} \) | 1.89.j |
| 97 | \( 1 - 16 T + p T^{2} \) | 1.97.aq |
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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
−13.61247696335165, −13.00164756914028, −12.54718274726506, −12.18046822463499, −11.66801062308505, −11.22754064839354, −10.43499488020285, −10.23408849051090, −9.737586818320568, −9.228944405774891, −8.857273501305820, −8.454951593629834, −7.331529323286654, −7.172001426132518, −6.547668174479450, −6.211745007859068, −5.855185771343464, −5.012419979837405, −4.444460239145612, −3.624250325214527, −3.252027283362187, −2.558984685199747, −1.682541696628268, −1.210253616481271, −0.3026096075521050,
0.3026096075521050, 1.210253616481271, 1.682541696628268, 2.558984685199747, 3.252027283362187, 3.624250325214527, 4.444460239145612, 5.012419979837405, 5.855185771343464, 6.211745007859068, 6.547668174479450, 7.172001426132518, 7.331529323286654, 8.454951593629834, 8.857273501305820, 9.228944405774891, 9.737586818320568, 10.23408849051090, 10.43499488020285, 11.22754064839354, 11.66801062308505, 12.18046822463499, 12.54718274726506, 13.00164756914028, 13.61247696335165
