| L(s) = 1 | − 2·5-s + 4·11-s − 2·13-s − 17-s + 8·23-s − 25-s + 6·29-s + 2·37-s − 10·41-s − 8·43-s − 8·47-s − 7·49-s + 10·53-s − 8·55-s + 10·61-s + 4·65-s − 8·67-s + 8·71-s − 6·73-s + 8·79-s + 2·85-s + 6·89-s + 10·97-s + 10·101-s + 8·103-s + 12·107-s + 2·109-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 1.20·11-s − 0.554·13-s − 0.242·17-s + 1.66·23-s − 1/5·25-s + 1.11·29-s + 0.328·37-s − 1.56·41-s − 1.21·43-s − 1.16·47-s − 49-s + 1.37·53-s − 1.07·55-s + 1.28·61-s + 0.496·65-s − 0.977·67-s + 0.949·71-s − 0.702·73-s + 0.900·79-s + 0.216·85-s + 0.635·89-s + 1.01·97-s + 0.995·101-s + 0.788·103-s + 1.16·107-s + 0.191·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.555517415\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.555517415\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 17 | \( 1 + T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 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
−8.384347073663336111830605588981, −7.46714939780246214553438635160, −6.83839094531278866839998104776, −6.33960263694689226714267518407, −5.09195372936571830243238085725, −4.61802838691997911032337704314, −3.67315677615395964382032547921, −3.08781219425605159013106321242, −1.83665648538448037277611840139, −0.69373224486184555123034995038,
0.69373224486184555123034995038, 1.83665648538448037277611840139, 3.08781219425605159013106321242, 3.67315677615395964382032547921, 4.61802838691997911032337704314, 5.09195372936571830243238085725, 6.33960263694689226714267518407, 6.83839094531278866839998104776, 7.46714939780246214553438635160, 8.384347073663336111830605588981
