| L(s) = 1 | − 2·4-s − 7-s − 7·13-s + 4·16-s + 2·28-s + 11·31-s + 11·37-s + 8·43-s − 6·49-s + 14·52-s + 14·61-s − 8·64-s − 16·67-s + 17·73-s + 17·79-s + 7·91-s + 5·97-s + 101-s + 103-s + 107-s + 109-s − 4·112-s + 113-s + ⋯ |
| L(s) = 1 | − 4-s − 0.377·7-s − 1.94·13-s + 16-s + 0.377·28-s + 1.97·31-s + 1.80·37-s + 1.21·43-s − 6/7·49-s + 1.94·52-s + 1.79·61-s − 64-s − 1.95·67-s + 1.98·73-s + 1.91·79-s + 0.733·91-s + 0.507·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s − 0.377·112-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.358525514\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.358525514\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 19 | \( 1 \) | |
| good | 2 | \( 1 + p T^{2} \) | 1.2.a |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 7 T + p T^{2} \) | 1.13.h |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 11 T + p T^{2} \) | 1.31.al |
| 37 | \( 1 - 11 T + p T^{2} \) | 1.37.al |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 + 16 T + p T^{2} \) | 1.67.q |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 17 T + p T^{2} \) | 1.73.ar |
| 79 | \( 1 - 17 T + p T^{2} \) | 1.79.ar |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 - 5 T + p T^{2} \) | 1.97.af |
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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
−13.95843099705540, −13.48141791272211, −12.97652726989220, −12.56017521489951, −12.04020947829901, −11.72851474210719, −10.87653310826973, −10.34255693446828, −9.826343243050814, −9.448347303376431, −9.222636209516570, −8.236038454687416, −8.010231294180176, −7.471462555422555, −6.756169924586633, −6.274902707170756, −5.552489280614041, −5.051892659704221, −4.479643157755427, −4.184598756906978, −3.283542234663235, −2.695879401841187, −2.188312684676967, −1.005884419280776, −0.4621899032503360,
0.4621899032503360, 1.005884419280776, 2.188312684676967, 2.695879401841187, 3.283542234663235, 4.184598756906978, 4.479643157755427, 5.051892659704221, 5.552489280614041, 6.274902707170756, 6.756169924586633, 7.471462555422555, 8.010231294180176, 8.236038454687416, 9.222636209516570, 9.448347303376431, 9.826343243050814, 10.34255693446828, 10.87653310826973, 11.72851474210719, 12.04020947829901, 12.56017521489951, 12.97652726989220, 13.48141791272211, 13.95843099705540
