| L(s) = 1 | − 2·4-s − 2·5-s + 3·11-s + 2·13-s + 4·16-s + 7·17-s − 19-s + 4·20-s − 5·23-s − 25-s − 2·29-s + 10·31-s + 8·37-s − 6·41-s + 12·43-s − 6·44-s + 5·47-s − 4·52-s − 4·53-s − 6·55-s − 14·59-s − 13·61-s − 8·64-s − 4·65-s − 2·67-s − 14·68-s + 10·71-s + ⋯ |
| L(s) = 1 | − 4-s − 0.894·5-s + 0.904·11-s + 0.554·13-s + 16-s + 1.69·17-s − 0.229·19-s + 0.894·20-s − 1.04·23-s − 1/5·25-s − 0.371·29-s + 1.79·31-s + 1.31·37-s − 0.937·41-s + 1.82·43-s − 0.904·44-s + 0.729·47-s − 0.554·52-s − 0.549·53-s − 0.809·55-s − 1.82·59-s − 1.66·61-s − 64-s − 0.496·65-s − 0.244·67-s − 1.69·68-s + 1.18·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8379 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8379 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.376961074\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.376961074\) |
| \(L(\frac{3}{2})\) |
|
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 | 3 | \( 1 \) | |
| 7 | \( 1 \) | |
| 19 | \( 1 + T \) | |
| good | 2 | \( 1 + p T^{2} \) | 1.2.a |
| 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 7 T + p T^{2} \) | 1.17.ah |
| 23 | \( 1 + 5 T + p T^{2} \) | 1.23.f |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 10 T + p T^{2} \) | 1.31.ak |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 - 5 T + p T^{2} \) | 1.47.af |
| 53 | \( 1 + 4 T + p T^{2} \) | 1.53.e |
| 59 | \( 1 + 14 T + p T^{2} \) | 1.59.o |
| 61 | \( 1 + 13 T + p T^{2} \) | 1.61.n |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 - 10 T + p T^{2} \) | 1.71.ak |
| 73 | \( 1 - T + p T^{2} \) | 1.73.ab |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 - 9 T + p T^{2} \) | 1.83.aj |
| 89 | \( 1 + 18 T + p T^{2} \) | 1.89.s |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
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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
−7.942112405215570553447451614100, −7.37808114110969091021327732103, −6.12036053632813551658285195805, −5.92964228599396174102198423298, −4.78247428706730706843854394928, −4.18272557043558835312270209740, −3.69983028069520210093884438358, −2.94181228695939632222766983601, −1.44883134860552069991703962127, −0.63963823563746829525639554786,
0.63963823563746829525639554786, 1.44883134860552069991703962127, 2.94181228695939632222766983601, 3.69983028069520210093884438358, 4.18272557043558835312270209740, 4.78247428706730706843854394928, 5.92964228599396174102198423298, 6.12036053632813551658285195805, 7.37808114110969091021327732103, 7.942112405215570553447451614100
