| L(s) = 1 | − 2-s + 3-s − 4-s − 6-s − 3·7-s + 3·8-s + 9-s − 5·11-s − 12-s + 3·14-s − 16-s + 2·17-s − 18-s + 6·19-s − 3·21-s + 5·22-s + 2·23-s + 3·24-s + 27-s + 3·28-s + 8·29-s − 10·31-s − 5·32-s − 5·33-s − 2·34-s − 36-s + 4·37-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.408·6-s − 1.13·7-s + 1.06·8-s + 1/3·9-s − 1.50·11-s − 0.288·12-s + 0.801·14-s − 1/4·16-s + 0.485·17-s − 0.235·18-s + 1.37·19-s − 0.654·21-s + 1.06·22-s + 0.417·23-s + 0.612·24-s + 0.192·27-s + 0.566·28-s + 1.48·29-s − 1.79·31-s − 0.883·32-s − 0.870·33-s − 0.342·34-s − 1/6·36-s + 0.657·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 138675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.393743543\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.393743543\) |
| \(L(\frac{3}{2})\) |
|
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 | 3 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 43 | \( 1 \) | |
| good | 2 | \( 1 + T + p T^{2} \) | 1.2.b |
| 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 11 | \( 1 + 5 T + p T^{2} \) | 1.11.f |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 - 8 T + p T^{2} \) | 1.29.ai |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 - 5 T + p T^{2} \) | 1.53.af |
| 59 | \( 1 + 13 T + p T^{2} \) | 1.59.n |
| 61 | \( 1 - 7 T + p T^{2} \) | 1.61.ah |
| 67 | \( 1 - 13 T + p T^{2} \) | 1.67.an |
| 71 | \( 1 - 9 T + p T^{2} \) | 1.71.aj |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 + 14 T + p T^{2} \) | 1.83.o |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 + 18 T + p T^{2} \) | 1.97.s |
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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
−13.40627858726744, −12.99833368345731, −12.53057055109997, −12.29174012637872, −11.15407322513359, −11.01065672820244, −10.23852056485821, −9.942145234894100, −9.542641348307839, −9.182197638408380, −8.600695437057137, −7.970805876350839, −7.775514479440926, −7.160924767598817, −6.783744401838618, −5.821014109278700, −5.468613789227744, −4.907066814637014, −4.275479555500660, −3.593205484673494, −3.066777163610055, −2.667678689213733, −1.866733982573054, −0.9502217791959563, −0.4793566495464841,
0.4793566495464841, 0.9502217791959563, 1.866733982573054, 2.667678689213733, 3.066777163610055, 3.593205484673494, 4.275479555500660, 4.907066814637014, 5.468613789227744, 5.821014109278700, 6.783744401838618, 7.160924767598817, 7.775514479440926, 7.970805876350839, 8.600695437057137, 9.182197638408380, 9.542641348307839, 9.942145234894100, 10.23852056485821, 11.01065672820244, 11.15407322513359, 12.29174012637872, 12.53057055109997, 12.99833368345731, 13.40627858726744
