| L(s) = 1 | + 5-s + 2·11-s − 2·13-s − 2·19-s + 8·23-s + 25-s − 29-s + 2·31-s − 4·37-s + 10·41-s + 4·43-s − 12·47-s − 7·49-s + 6·53-s + 2·55-s + 12·59-s − 10·61-s − 2·65-s + 12·67-s − 12·71-s + 12·73-s + 2·79-s + 4·83-s + 10·89-s − 2·95-s + 8·97-s + 6·101-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.603·11-s − 0.554·13-s − 0.458·19-s + 1.66·23-s + 1/5·25-s − 0.185·29-s + 0.359·31-s − 0.657·37-s + 1.56·41-s + 0.609·43-s − 1.75·47-s − 49-s + 0.824·53-s + 0.269·55-s + 1.56·59-s − 1.28·61-s − 0.248·65-s + 1.46·67-s − 1.42·71-s + 1.40·73-s + 0.225·79-s + 0.439·83-s + 1.05·89-s − 0.205·95-s + 0.812·97-s + 0.597·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5220 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.200375115\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.200375115\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 29 | \( 1 + T \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 - 12 T + p T^{2} \) | 1.73.am |
| 79 | \( 1 - 2 T + p T^{2} \) | 1.79.ac |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 - 8 T + p T^{2} \) | 1.97.ai |
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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
−8.250401784634935116424626951763, −7.40033147493330085989839512369, −6.74495336694957776906239452941, −6.14396794113848450870733092738, −5.21459022786390552201057580365, −4.65076926295003452163252799768, −3.67639923117339100925767909920, −2.80202439488109963705600728719, −1.90973723859044679527840059360, −0.818507504315089756918734201418,
0.818507504315089756918734201418, 1.90973723859044679527840059360, 2.80202439488109963705600728719, 3.67639923117339100925767909920, 4.65076926295003452163252799768, 5.21459022786390552201057580365, 6.14396794113848450870733092738, 6.74495336694957776906239452941, 7.40033147493330085989839512369, 8.250401784634935116424626951763
