| L(s) = 1 | − 2-s + 4-s + 5-s − 2·7-s − 8-s − 10-s − 3·11-s + 2·14-s + 16-s − 2·17-s + 20-s + 3·22-s + 23-s − 4·25-s − 2·28-s − 2·29-s − 5·31-s − 32-s + 2·34-s − 2·35-s − 7·37-s − 40-s − 7·41-s + 8·43-s − 3·44-s − 46-s + 13·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.755·7-s − 0.353·8-s − 0.316·10-s − 0.904·11-s + 0.534·14-s + 1/4·16-s − 0.485·17-s + 0.223·20-s + 0.639·22-s + 0.208·23-s − 4/5·25-s − 0.377·28-s − 0.371·29-s − 0.898·31-s − 0.176·32-s + 0.342·34-s − 0.338·35-s − 1.15·37-s − 0.158·40-s − 1.09·41-s + 1.21·43-s − 0.452·44-s − 0.147·46-s + 1.89·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 149454 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 149454 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4716196275\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4716196275\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 19 | \( 1 \) | |
| 23 | \( 1 - T \) | |
| good | 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 5 T + p T^{2} \) | 1.31.f |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 41 | \( 1 + 7 T + p T^{2} \) | 1.41.h |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 - 13 T + p T^{2} \) | 1.47.an |
| 53 | \( 1 + 9 T + p T^{2} \) | 1.53.j |
| 59 | \( 1 - 14 T + p T^{2} \) | 1.59.ao |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 - 5 T + p T^{2} \) | 1.71.af |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 15 T + p T^{2} \) | 1.83.p |
| 89 | \( 1 - 8 T + p T^{2} \) | 1.89.ai |
| 97 | \( 1 + p T^{2} \) | 1.97.a |
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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
−13.24727862589298, −12.97877715879841, −12.32200457488849, −12.00163924083125, −11.20139416266078, −10.90325915996794, −10.43080637246551, −9.857710308361170, −9.603641015288010, −9.061757552733868, −8.545639272015868, −8.081025264771394, −7.500050632703076, −6.881400525243445, −6.730618838469522, −5.870629573903461, −5.498276972802190, −5.110243895474437, −4.096145928376281, −3.703218329360723, −2.970648423283987, −2.369548166671795, −1.983012849151041, −1.135891415294210, −0.2351914912300634,
0.2351914912300634, 1.135891415294210, 1.983012849151041, 2.369548166671795, 2.970648423283987, 3.703218329360723, 4.096145928376281, 5.110243895474437, 5.498276972802190, 5.870629573903461, 6.730618838469522, 6.881400525243445, 7.500050632703076, 8.081025264771394, 8.545639272015868, 9.061757552733868, 9.603641015288010, 9.857710308361170, 10.43080637246551, 10.90325915996794, 11.20139416266078, 12.00163924083125, 12.32200457488849, 12.97877715879841, 13.24727862589298
