| L(s) = 1 | + 5-s + 13-s + 6·17-s − 4·19-s + 25-s + 2·29-s − 2·37-s + 2·41-s + 4·43-s + 4·47-s − 7·49-s + 10·53-s + 8·59-s − 2·61-s + 65-s + 4·67-s − 12·71-s − 6·73-s − 16·83-s + 6·85-s + 10·89-s − 4·95-s + 2·97-s + 10·101-s + 8·103-s + 12·107-s + 6·109-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.277·13-s + 1.45·17-s − 0.917·19-s + 1/5·25-s + 0.371·29-s − 0.328·37-s + 0.312·41-s + 0.609·43-s + 0.583·47-s − 49-s + 1.37·53-s + 1.04·59-s − 0.256·61-s + 0.124·65-s + 0.488·67-s − 1.42·71-s − 0.702·73-s − 1.75·83-s + 0.650·85-s + 1.05·89-s − 0.410·95-s + 0.203·97-s + 0.995·101-s + 0.788·103-s + 1.16·107-s + 0.574·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.396438733\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.396438733\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 13 | \( 1 - T \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 16 T + p T^{2} \) | 1.83.q |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−7.62324378299215902741062758837, −7.09785396200064465214797516695, −6.17979865503199723019379117152, −5.78207483051557126459637646253, −5.00212344206855813891710351138, −4.19941579765160930774814843783, −3.42136055860817229007355967254, −2.61219362346689852167676333129, −1.71666572186115290853477084758, −0.76321534940383283477320476557,
0.76321534940383283477320476557, 1.71666572186115290853477084758, 2.61219362346689852167676333129, 3.42136055860817229007355967254, 4.19941579765160930774814843783, 5.00212344206855813891710351138, 5.78207483051557126459637646253, 6.17979865503199723019379117152, 7.09785396200064465214797516695, 7.62324378299215902741062758837
