| L(s) = 1 | − 2·2-s + 3-s + 2·4-s − 5-s − 2·6-s + 7-s + 9-s + 2·10-s + 3·11-s + 2·12-s − 6·13-s − 2·14-s − 15-s − 4·16-s − 2·18-s − 19-s − 2·20-s + 21-s − 6·22-s + 25-s + 12·26-s + 27-s + 2·28-s − 5·29-s + 2·30-s − 8·31-s + 8·32-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 0.577·3-s + 4-s − 0.447·5-s − 0.816·6-s + 0.377·7-s + 1/3·9-s + 0.632·10-s + 0.904·11-s + 0.577·12-s − 1.66·13-s − 0.534·14-s − 0.258·15-s − 16-s − 0.471·18-s − 0.229·19-s − 0.447·20-s + 0.218·21-s − 1.27·22-s + 1/5·25-s + 2.35·26-s + 0.192·27-s + 0.377·28-s − 0.928·29-s + 0.365·30-s − 1.43·31-s + 1.41·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30345 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30345 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) | |
| 5 | \( 1 + T \) | |
| 7 | \( 1 - T \) | |
| 17 | \( 1 \) | |
| good | 2 | \( 1 + p T + p T^{2} \) | 1.2.c |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 5 T + p T^{2} \) | 1.29.f |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 5 T + p T^{2} \) | 1.41.af |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 4 T + p T^{2} \) | 1.53.e |
| 59 | \( 1 - 9 T + p T^{2} \) | 1.59.aj |
| 61 | \( 1 + 11 T + p T^{2} \) | 1.61.l |
| 67 | \( 1 + 6 T + p T^{2} \) | 1.67.g |
| 71 | \( 1 - 15 T + p T^{2} \) | 1.71.ap |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 - T + p T^{2} \) | 1.79.ab |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + 9 T + p T^{2} \) | 1.89.j |
| 97 | \( 1 - 4 T + p T^{2} \) | 1.97.ae |
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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
−15.39815908733262, −14.92334688762954, −14.29661988403789, −14.19986695967603, −13.15270342136692, −12.67764787669202, −12.08663694386909, −11.47528535443602, −11.00196840064523, −10.45957801423225, −9.750713552145864, −9.302711620655586, −9.084185270681167, −8.297393940301344, −7.838301421815433, −7.226968467923554, −7.115405668586070, −6.157005933389037, −5.264032807722800, −4.525115315483028, −4.044003857501402, −3.188885951242071, −2.249484499354134, −1.853975503732718, −0.8987737127439721, 0,
0.8987737127439721, 1.853975503732718, 2.249484499354134, 3.188885951242071, 4.044003857501402, 4.525115315483028, 5.264032807722800, 6.157005933389037, 7.115405668586070, 7.226968467923554, 7.838301421815433, 8.297393940301344, 9.084185270681167, 9.302711620655586, 9.750713552145864, 10.45957801423225, 11.00196840064523, 11.47528535443602, 12.08663694386909, 12.67764787669202, 13.15270342136692, 14.19986695967603, 14.29661988403789, 14.92334688762954, 15.39815908733262
