| L(s) = 1 | − 3·9-s + 11-s − 2·13-s − 4·17-s + 2·19-s + 5·23-s + 29-s + 2·31-s + 3·37-s − 12·41-s + 11·43-s − 2·47-s + 6·53-s + 10·59-s − 4·61-s + 67-s − 3·71-s − 9·79-s + 9·81-s + 2·83-s + 6·89-s − 14·97-s − 3·99-s − 12·101-s − 14·103-s − 12·107-s + 5·109-s + ⋯ |
| L(s) = 1 | − 9-s + 0.301·11-s − 0.554·13-s − 0.970·17-s + 0.458·19-s + 1.04·23-s + 0.185·29-s + 0.359·31-s + 0.493·37-s − 1.87·41-s + 1.67·43-s − 0.291·47-s + 0.824·53-s + 1.30·59-s − 0.512·61-s + 0.122·67-s − 0.356·71-s − 1.01·79-s + 81-s + 0.219·83-s + 0.635·89-s − 1.42·97-s − 0.301·99-s − 1.19·101-s − 1.37·103-s − 1.16·107-s + 0.478·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 11 | \( 1 - T + p T^{2} \) | 1.11.ab |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - 5 T + p T^{2} \) | 1.23.af |
| 29 | \( 1 - T + p T^{2} \) | 1.29.ab |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 - 3 T + p T^{2} \) | 1.37.ad |
| 41 | \( 1 + 12 T + p T^{2} \) | 1.41.m |
| 43 | \( 1 - 11 T + p T^{2} \) | 1.43.al |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 10 T + p T^{2} \) | 1.59.ak |
| 61 | \( 1 + 4 T + p T^{2} \) | 1.61.e |
| 67 | \( 1 - T + p T^{2} \) | 1.67.ab |
| 71 | \( 1 + 3 T + p T^{2} \) | 1.71.d |
| 73 | \( 1 + p T^{2} \) | 1.73.a |
| 79 | \( 1 + 9 T + p T^{2} \) | 1.79.j |
| 83 | \( 1 - 2 T + p T^{2} \) | 1.83.ac |
| 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
−7.17207091127291452320332542531, −6.77177893835369951330388294774, −5.92797423982630294533139924800, −5.28426993966509160103282548343, −4.62894610657374888576397068988, −3.79453082044466306493126778703, −2.89019816560611373046818106532, −2.35856563985089138356402756254, −1.15363929961997794187784404159, 0,
1.15363929961997794187784404159, 2.35856563985089138356402756254, 2.89019816560611373046818106532, 3.79453082044466306493126778703, 4.62894610657374888576397068988, 5.28426993966509160103282548343, 5.92797423982630294533139924800, 6.77177893835369951330388294774, 7.17207091127291452320332542531
