| L(s) = 1 | − 2·4-s + 4·13-s + 4·16-s − 17-s − 2·19-s − 9·23-s − 5·25-s + 9·29-s − 5·31-s + 2·37-s − 43-s − 9·47-s − 8·52-s + 9·59-s + 61-s − 8·64-s + 5·67-s + 2·68-s + 9·71-s + 7·73-s + 4·76-s − 10·79-s − 9·89-s + 18·92-s + 97-s + 10·100-s + 101-s + ⋯ |
| L(s) = 1 | − 4-s + 1.10·13-s + 16-s − 0.242·17-s − 0.458·19-s − 1.87·23-s − 25-s + 1.67·29-s − 0.898·31-s + 0.328·37-s − 0.152·43-s − 1.31·47-s − 1.10·52-s + 1.17·59-s + 0.128·61-s − 64-s + 0.610·67-s + 0.242·68-s + 1.06·71-s + 0.819·73-s + 0.458·76-s − 1.12·79-s − 0.953·89-s + 1.87·92-s + 0.101·97-s + 100-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22491 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22491 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.160658565\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.160658565\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 17 | \( 1 + T \) | |
| good | 2 | \( 1 + p T^{2} \) | 1.2.a |
| 5 | \( 1 + p T^{2} \) | 1.5.a |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + 9 T + p T^{2} \) | 1.23.j |
| 29 | \( 1 - 9 T + p T^{2} \) | 1.29.aj |
| 31 | \( 1 + 5 T + p T^{2} \) | 1.31.f |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 + 9 T + p T^{2} \) | 1.47.j |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 - 9 T + p T^{2} \) | 1.59.aj |
| 61 | \( 1 - T + p T^{2} \) | 1.61.ab |
| 67 | \( 1 - 5 T + p T^{2} \) | 1.67.af |
| 71 | \( 1 - 9 T + p T^{2} \) | 1.71.aj |
| 73 | \( 1 - 7 T + p T^{2} \) | 1.73.ah |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 9 T + p T^{2} \) | 1.89.j |
| 97 | \( 1 - T + p T^{2} \) | 1.97.ab |
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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
−15.70334439434521, −14.81888111487961, −14.34021056817549, −13.81426783207422, −13.47377734071485, −12.85774790722559, −12.33998026730250, −11.73888927350380, −11.11991798151383, −10.49456898709483, −9.810079878071586, −9.615062916864407, −8.653239960076703, −8.289496604503061, −7.986300978405577, −6.985155263905476, −6.277250902682972, −5.820223812737499, −5.144703446083190, −4.323188308050148, −3.938008174073500, −3.317476207745994, −2.260154188607823, −1.460395707393147, −0.4552846891636460,
0.4552846891636460, 1.460395707393147, 2.260154188607823, 3.317476207745994, 3.938008174073500, 4.323188308050148, 5.144703446083190, 5.820223812737499, 6.277250902682972, 6.985155263905476, 7.986300978405577, 8.289496604503061, 8.653239960076703, 9.615062916864407, 9.810079878071586, 10.49456898709483, 11.11991798151383, 11.73888927350380, 12.33998026730250, 12.85774790722559, 13.47377734071485, 13.81426783207422, 14.34021056817549, 14.81888111487961, 15.70334439434521
