| L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s + 7-s + 8-s + 9-s − 10-s + 11-s + 12-s + 14-s − 15-s + 16-s − 3·17-s + 18-s + 6·19-s − 20-s + 21-s + 22-s + 3·23-s + 24-s − 4·25-s + 27-s + 28-s + 9·29-s − 30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.301·11-s + 0.288·12-s + 0.267·14-s − 0.258·15-s + 1/4·16-s − 0.727·17-s + 0.235·18-s + 1.37·19-s − 0.223·20-s + 0.218·21-s + 0.213·22-s + 0.625·23-s + 0.204·24-s − 4/5·25-s + 0.192·27-s + 0.188·28-s + 1.67·29-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 157146 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 157146 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 11 | \( 1 - T \) | |
| 2381 | \( 1 - T \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 - 9 T + p T^{2} \) | 1.29.aj |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 11 T + p T^{2} \) | 1.37.l |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 9 T + p T^{2} \) | 1.53.j |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 - 5 T + p T^{2} \) | 1.67.af |
| 71 | \( 1 - 9 T + p T^{2} \) | 1.71.aj |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + 11 T + p T^{2} \) | 1.79.l |
| 83 | \( 1 - 14 T + p T^{2} \) | 1.83.ao |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−13.56925898214998, −13.27578265333728, −12.46917503369775, −12.19881146192407, −11.67336232585533, −11.35013526148864, −10.74332651868979, −10.26804354007225, −9.647995739112338, −9.267192118134361, −8.583987289437881, −8.124843580342930, −7.801665432207897, −7.120579765513496, −6.671321251742180, −6.320692831160989, −5.403847338949345, −5.033923647531729, −4.510286426682700, −4.031627127580779, −3.286012048648276, −3.067473678947333, −2.338165568745997, −1.576858747098341, −1.113415696490421, 0,
1.113415696490421, 1.576858747098341, 2.338165568745997, 3.067473678947333, 3.286012048648276, 4.031627127580779, 4.510286426682700, 5.033923647531729, 5.403847338949345, 6.320692831160989, 6.671321251742180, 7.120579765513496, 7.801665432207897, 8.124843580342930, 8.583987289437881, 9.267192118134361, 9.647995739112338, 10.26804354007225, 10.74332651868979, 11.35013526148864, 11.67336232585533, 12.19881146192407, 12.46917503369775, 13.27578265333728, 13.56925898214998
