| L(s) = 1 | − 3-s − 3·5-s + 2·7-s − 2·9-s − 3·11-s − 5·13-s + 3·15-s − 4·17-s − 2·21-s + 4·25-s + 5·27-s − 29-s + 9·31-s + 3·33-s − 6·35-s + 8·37-s + 5·39-s − 2·41-s − 11·43-s + 6·45-s − 7·47-s − 3·49-s + 4·51-s + 9·53-s + 9·55-s + 4·59-s − 12·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.34·5-s + 0.755·7-s − 2/3·9-s − 0.904·11-s − 1.38·13-s + 0.774·15-s − 0.970·17-s − 0.436·21-s + 4/5·25-s + 0.962·27-s − 0.185·29-s + 1.61·31-s + 0.522·33-s − 1.01·35-s + 1.31·37-s + 0.800·39-s − 0.312·41-s − 1.67·43-s + 0.894·45-s − 1.02·47-s − 3/7·49-s + 0.560·51-s + 1.23·53-s + 1.21·55-s + 0.520·59-s − 1.53·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 29 | \( 1 + T \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 + 5 T + p T^{2} \) | 1.13.f |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 31 | \( 1 - 9 T + p T^{2} \) | 1.31.aj |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 11 T + p T^{2} \) | 1.43.l |
| 47 | \( 1 + 7 T + p T^{2} \) | 1.47.h |
| 53 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 12 T + p T^{2} \) | 1.61.m |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 - 2 T + p T^{2} \) | 1.71.ac |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 - 3 T + p T^{2} \) | 1.79.ad |
| 83 | \( 1 + 16 T + p T^{2} \) | 1.83.q |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 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
−11.60564917575106526236403888893, −11.10522087321152350977566130714, −9.963920685863865440792238727374, −8.410269641065470703778598717182, −7.86451956272498755707499366016, −6.72783331557161674988315423172, −5.19107469246100923800577898520, −4.45944873695965118173142297517, −2.72675929619515706873392619053, 0,
2.72675929619515706873392619053, 4.45944873695965118173142297517, 5.19107469246100923800577898520, 6.72783331557161674988315423172, 7.86451956272498755707499366016, 8.410269641065470703778598717182, 9.963920685863865440792238727374, 11.10522087321152350977566130714, 11.60564917575106526236403888893
