| L(s) = 1 | − 7-s + 3·11-s + 13-s − 7·17-s − 6·19-s − 5·25-s − 29-s − 6·31-s + 2·37-s + 10·41-s + 10·43-s − 47-s − 6·49-s − 6·53-s + 10·59-s − 12·61-s − 3·67-s + 14·71-s − 16·73-s − 3·77-s − 12·79-s − 16·83-s − 13·89-s − 91-s + 10·97-s − 3·101-s − 16·103-s + ⋯ |
| L(s) = 1 | − 0.377·7-s + 0.904·11-s + 0.277·13-s − 1.69·17-s − 1.37·19-s − 25-s − 0.185·29-s − 1.07·31-s + 0.328·37-s + 1.56·41-s + 1.52·43-s − 0.145·47-s − 6/7·49-s − 0.824·53-s + 1.30·59-s − 1.53·61-s − 0.366·67-s + 1.66·71-s − 1.87·73-s − 0.341·77-s − 1.35·79-s − 1.75·83-s − 1.37·89-s − 0.104·91-s + 1.01·97-s − 0.298·101-s − 1.57·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2088 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2088 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 29 | \( 1 + T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 17 | \( 1 + 7 T + p T^{2} \) | 1.17.h |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 31 | \( 1 + 6 T + p T^{2} \) | 1.31.g |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 47 | \( 1 + T + p T^{2} \) | 1.47.b |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 10 T + p T^{2} \) | 1.59.ak |
| 61 | \( 1 + 12 T + p T^{2} \) | 1.61.m |
| 67 | \( 1 + 3 T + p T^{2} \) | 1.67.d |
| 71 | \( 1 - 14 T + p T^{2} \) | 1.71.ao |
| 73 | \( 1 + 16 T + p T^{2} \) | 1.73.q |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 + 16 T + p T^{2} \) | 1.83.q |
| 89 | \( 1 + 13 T + p T^{2} \) | 1.89.n |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−8.961564798926642365518199827035, −7.987444335389149297704049163927, −7.04396284671871403679667281477, −6.35498380067665774619185118042, −5.75391290995307700430504967677, −4.32441970583018028952972985141, −4.03281536182273214248961485688, −2.67214543045075489567581310934, −1.68572189637400302777569607414, 0,
1.68572189637400302777569607414, 2.67214543045075489567581310934, 4.03281536182273214248961485688, 4.32441970583018028952972985141, 5.75391290995307700430504967677, 6.35498380067665774619185118042, 7.04396284671871403679667281477, 7.987444335389149297704049163927, 8.961564798926642365518199827035
