| L(s) = 1 | + 3-s − 2·5-s + 9-s + 2·11-s − 6·13-s − 2·15-s − 2·17-s + 19-s + 6·23-s − 25-s + 27-s + 8·29-s + 8·31-s + 2·33-s + 10·37-s − 6·39-s − 8·41-s + 8·43-s − 2·45-s − 2·47-s − 2·51-s + 8·53-s − 4·55-s + 57-s − 6·61-s + 12·65-s + 4·67-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s + 1/3·9-s + 0.603·11-s − 1.66·13-s − 0.516·15-s − 0.485·17-s + 0.229·19-s + 1.25·23-s − 1/5·25-s + 0.192·27-s + 1.48·29-s + 1.43·31-s + 0.348·33-s + 1.64·37-s − 0.960·39-s − 1.24·41-s + 1.21·43-s − 0.298·45-s − 0.291·47-s − 0.280·51-s + 1.09·53-s − 0.539·55-s + 0.132·57-s − 0.768·61-s + 1.48·65-s + 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 178752 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 178752 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.052760165\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.052760165\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| 19 | \( 1 - T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 - 8 T + p T^{2} \) | 1.29.ai |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 + 8 T + p T^{2} \) | 1.41.i |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 - 8 T + p T^{2} \) | 1.53.ai |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 14 T + p T^{2} \) | 1.83.ao |
| 89 | \( 1 - 16 T + p T^{2} \) | 1.89.aq |
| 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
−13.10006366486492, −12.74515953395367, −12.01751122870116, −11.87550790946628, −11.51553445943110, −10.74025100704321, −10.30035999872577, −9.788425376527666, −9.355418722560683, −8.838594136247375, −8.401311466130976, −7.783414997196470, −7.510462693169279, −6.972107628131173, −6.503860908138196, −5.960129283694586, −4.987888403912724, −4.673096945319140, −4.354243190601931, −3.598027933490033, −3.052892662224067, −2.553694262655740, −2.044028033638802, −0.9836232035965772, −0.5807692008066283,
0.5807692008066283, 0.9836232035965772, 2.044028033638802, 2.553694262655740, 3.052892662224067, 3.598027933490033, 4.354243190601931, 4.673096945319140, 4.987888403912724, 5.960129283694586, 6.503860908138196, 6.972107628131173, 7.510462693169279, 7.783414997196470, 8.401311466130976, 8.838594136247375, 9.355418722560683, 9.788425376527666, 10.30035999872577, 10.74025100704321, 11.51553445943110, 11.87550790946628, 12.01751122870116, 12.74515953395367, 13.10006366486492
