| L(s) = 1 | + 5-s − 2·11-s + 3·13-s + 2·19-s + 23-s − 4·25-s + 5·29-s + 37-s − 3·41-s − 7·43-s − 13·47-s + 6·53-s − 2·55-s + 12·59-s − 4·61-s + 3·65-s + 8·71-s + 2·73-s − 10·79-s + 4·83-s + 16·89-s + 2·95-s − 11·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.603·11-s + 0.832·13-s + 0.458·19-s + 0.208·23-s − 4/5·25-s + 0.928·29-s + 0.164·37-s − 0.468·41-s − 1.06·43-s − 1.89·47-s + 0.824·53-s − 0.269·55-s + 1.56·59-s − 0.512·61-s + 0.372·65-s + 0.949·71-s + 0.234·73-s − 1.12·79-s + 0.439·83-s + 1.69·89-s + 0.205·95-s − 1.11·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 3 | \( 1 \) | |
| 7 | \( 1 \) | |
| 23 | \( 1 - T \) | |
| good | 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 - 3 T + p T^{2} \) | 1.13.ad |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 29 | \( 1 - 5 T + p T^{2} \) | 1.29.af |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - T + p T^{2} \) | 1.37.ab |
| 41 | \( 1 + 3 T + p T^{2} \) | 1.41.d |
| 43 | \( 1 + 7 T + p T^{2} \) | 1.43.h |
| 47 | \( 1 + 13 T + p T^{2} \) | 1.47.n |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + 4 T + p T^{2} \) | 1.61.e |
| 67 | \( 1 + p T^{2} \) | 1.67.a |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 16 T + p T^{2} \) | 1.89.aq |
| 97 | \( 1 + 11 T + p T^{2} \) | 1.97.l |
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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
−14.13015485663339, −13.64303385221474, −13.24830404881875, −12.97041040118731, −12.18076853046270, −11.70444811201723, −11.32472139717617, −10.64897184405027, −10.22238890142951, −9.771602702428975, −9.278487283837959, −8.585275818613478, −8.157634858165976, −7.781466001569121, −6.868055150696699, −6.620696033216328, −5.943546999404804, −5.357907830140646, −4.998643343332859, −4.218174782824629, −3.575931543138115, −3.031584678527838, −2.336911654685973, −1.654215346072374, −0.9793917756413183, 0,
0.9793917756413183, 1.654215346072374, 2.336911654685973, 3.031584678527838, 3.575931543138115, 4.218174782824629, 4.998643343332859, 5.357907830140646, 5.943546999404804, 6.620696033216328, 6.868055150696699, 7.781466001569121, 8.157634858165976, 8.585275818613478, 9.278487283837959, 9.771602702428975, 10.22238890142951, 10.64897184405027, 11.32472139717617, 11.70444811201723, 12.18076853046270, 12.97041040118731, 13.24830404881875, 13.64303385221474, 14.13015485663339
