| L(s) = 1 | − 2·7-s − 4·11-s + 6·13-s + 2·17-s − 8·19-s + 6·23-s + 2·29-s − 4·31-s − 2·37-s + 10·41-s − 2·43-s + 2·47-s − 3·49-s + 2·53-s + 2·61-s − 6·67-s − 12·71-s − 10·73-s + 8·77-s + 8·79-s + 10·83-s + 6·89-s − 12·91-s − 10·97-s − 14·101-s + 2·103-s + 6·107-s + ⋯ |
| L(s) = 1 | − 0.755·7-s − 1.20·11-s + 1.66·13-s + 0.485·17-s − 1.83·19-s + 1.25·23-s + 0.371·29-s − 0.718·31-s − 0.328·37-s + 1.56·41-s − 0.304·43-s + 0.291·47-s − 3/7·49-s + 0.274·53-s + 0.256·61-s − 0.733·67-s − 1.42·71-s − 1.17·73-s + 0.911·77-s + 0.900·79-s + 1.09·83-s + 0.635·89-s − 1.25·91-s − 1.01·97-s − 1.39·101-s + 0.197·103-s + 0.580·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 - 2 T + p T^{2} \) | 1.47.ac |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 6 T + p T^{2} \) | 1.67.g |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 10 T + p T^{2} \) | 1.83.ak |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 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
−7.59610758444242161617177695712, −6.80268842085087504495737842973, −6.14500504390207752739946219388, −5.62849850723599259860957487685, −4.70127355445145585591695872396, −3.87860038693336130834403016662, −3.14992765442791720716852216804, −2.38947599972916053051339769753, −1.22044737534751131798568315114, 0,
1.22044737534751131798568315114, 2.38947599972916053051339769753, 3.14992765442791720716852216804, 3.87860038693336130834403016662, 4.70127355445145585591695872396, 5.62849850723599259860957487685, 6.14500504390207752739946219388, 6.80268842085087504495737842973, 7.59610758444242161617177695712
