| L(s) = 1 | + 5-s − 2·7-s + 2·13-s − 3·17-s − 5·19-s − 3·23-s + 25-s − 6·29-s − 5·31-s − 2·35-s + 2·37-s + 12·41-s − 8·43-s + 12·47-s − 3·49-s − 3·53-s − 6·59-s − 7·61-s + 2·65-s − 2·67-s − 12·71-s − 16·73-s + 79-s + 15·83-s − 3·85-s − 12·89-s − 4·91-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.755·7-s + 0.554·13-s − 0.727·17-s − 1.14·19-s − 0.625·23-s + 1/5·25-s − 1.11·29-s − 0.898·31-s − 0.338·35-s + 0.328·37-s + 1.87·41-s − 1.21·43-s + 1.75·47-s − 3/7·49-s − 0.412·53-s − 0.781·59-s − 0.896·61-s + 0.248·65-s − 0.244·67-s − 1.42·71-s − 1.87·73-s + 0.112·79-s + 1.64·83-s − 0.325·85-s − 1.27·89-s − 0.419·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 + 5 T + p T^{2} \) | 1.19.f |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 5 T + p T^{2} \) | 1.31.f |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 12 T + p T^{2} \) | 1.41.am |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 + 3 T + p T^{2} \) | 1.53.d |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 + 7 T + p T^{2} \) | 1.61.h |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 16 T + p T^{2} \) | 1.73.q |
| 79 | \( 1 - T + p T^{2} \) | 1.79.ab |
| 83 | \( 1 - 15 T + p T^{2} \) | 1.83.ap |
| 89 | \( 1 + 12 T + p T^{2} \) | 1.89.m |
| 97 | \( 1 + 16 T + p T^{2} \) | 1.97.q |
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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
−8.974323791517727613156969717226, −7.900894106986642924656260802762, −7.06762267986217976292626373786, −6.16509796689029791039542762193, −5.81567551008123256451318638521, −4.52615608168038150770349947310, −3.77514119356411728518356442270, −2.68476537451454374825128175456, −1.69695413361885578786474643023, 0,
1.69695413361885578786474643023, 2.68476537451454374825128175456, 3.77514119356411728518356442270, 4.52615608168038150770349947310, 5.81567551008123256451318638521, 6.16509796689029791039542762193, 7.06762267986217976292626373786, 7.900894106986642924656260802762, 8.974323791517727613156969717226
