| L(s) = 1 | − 2·7-s − 13-s + 6·17-s − 2·19-s − 23-s − 5·25-s + 3·29-s − 5·31-s + 8·37-s − 3·41-s − 8·43-s + 9·47-s − 3·49-s − 6·53-s − 12·59-s + 14·61-s − 8·67-s − 15·71-s − 7·73-s + 10·79-s + 6·83-s + 2·91-s − 10·97-s − 6·101-s − 2·103-s − 6·107-s − 16·109-s + ⋯ |
| L(s) = 1 | − 0.755·7-s − 0.277·13-s + 1.45·17-s − 0.458·19-s − 0.208·23-s − 25-s + 0.557·29-s − 0.898·31-s + 1.31·37-s − 0.468·41-s − 1.21·43-s + 1.31·47-s − 3/7·49-s − 0.824·53-s − 1.56·59-s + 1.79·61-s − 0.977·67-s − 1.78·71-s − 0.819·73-s + 1.12·79-s + 0.658·83-s + 0.209·91-s − 1.01·97-s − 0.597·101-s − 0.197·103-s − 0.580·107-s − 1.53·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 23 | \( 1 + T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 + 5 T + p T^{2} \) | 1.31.f |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 + 3 T + p T^{2} \) | 1.41.d |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 - 9 T + p T^{2} \) | 1.47.aj |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + 15 T + p T^{2} \) | 1.71.p |
| 73 | \( 1 + 7 T + p T^{2} \) | 1.73.h |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−8.120789330494647916179075071888, −7.61666438206331502600378578279, −6.71709540247988498373606792635, −6.00208974468012626810377293641, −5.33041149115783992949443030802, −4.28682971351670616728048423326, −3.46966194507991808988690503390, −2.66576538207574184886131321178, −1.45014338606706994297913460608, 0,
1.45014338606706994297913460608, 2.66576538207574184886131321178, 3.46966194507991808988690503390, 4.28682971351670616728048423326, 5.33041149115783992949443030802, 6.00208974468012626810377293641, 6.71709540247988498373606792635, 7.61666438206331502600378578279, 8.120789330494647916179075071888
