| L(s) = 1 | − 3-s − 2·4-s + 5-s − 3·7-s − 2·9-s − 5·11-s + 2·12-s + 4·13-s − 15-s + 4·16-s − 4·17-s − 8·19-s − 2·20-s + 3·21-s + 4·23-s + 25-s + 5·27-s + 6·28-s + 4·29-s + 2·31-s + 5·33-s − 3·35-s + 4·36-s + 37-s − 4·39-s − 5·41-s − 6·43-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 4-s + 0.447·5-s − 1.13·7-s − 2/3·9-s − 1.50·11-s + 0.577·12-s + 1.10·13-s − 0.258·15-s + 16-s − 0.970·17-s − 1.83·19-s − 0.447·20-s + 0.654·21-s + 0.834·23-s + 1/5·25-s + 0.962·27-s + 1.13·28-s + 0.742·29-s + 0.359·31-s + 0.870·33-s − 0.507·35-s + 2/3·36-s + 0.164·37-s − 0.640·39-s − 0.780·41-s − 0.914·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) | |
| 37 | \( 1 - T \) | |
| good | 2 | \( 1 + p T^{2} \) | 1.2.a |
| 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 11 | \( 1 + 5 T + p T^{2} \) | 1.11.f |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 41 | \( 1 + 5 T + p T^{2} \) | 1.41.f |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 - 9 T + p T^{2} \) | 1.47.aj |
| 53 | \( 1 - 3 T + p T^{2} \) | 1.53.ad |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 5 T + p T^{2} \) | 1.71.af |
| 73 | \( 1 + 15 T + p T^{2} \) | 1.73.p |
| 79 | \( 1 + 14 T + p T^{2} \) | 1.79.o |
| 83 | \( 1 - 11 T + p T^{2} \) | 1.83.al |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−12.39360295919822087145266861758, −10.82163125080847157281668664466, −10.33004705055258350495470777335, −8.998244863867538043901984087402, −8.371026417403938974286449500574, −6.53964005793593991571968990267, −5.74480366916894316991458613686, −4.56875092816775540121130748594, −2.94510070475246635819295308496, 0,
2.94510070475246635819295308496, 4.56875092816775540121130748594, 5.74480366916894316991458613686, 6.53964005793593991571968990267, 8.371026417403938974286449500574, 8.998244863867538043901984087402, 10.33004705055258350495470777335, 10.82163125080847157281668664466, 12.39360295919822087145266861758
