| L(s) = 1 | + 3-s + 2·5-s + 9-s − 11-s − 6·13-s + 2·15-s − 2·17-s − 4·19-s − 4·23-s − 25-s + 27-s − 6·29-s − 33-s − 6·37-s − 6·39-s + 6·41-s + 4·43-s + 2·45-s − 12·47-s − 2·51-s − 2·53-s − 2·55-s − 4·57-s − 12·59-s − 14·61-s − 12·65-s + 4·67-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894·5-s + 1/3·9-s − 0.301·11-s − 1.66·13-s + 0.516·15-s − 0.485·17-s − 0.917·19-s − 0.834·23-s − 1/5·25-s + 0.192·27-s − 1.11·29-s − 0.174·33-s − 0.986·37-s − 0.960·39-s + 0.937·41-s + 0.609·43-s + 0.298·45-s − 1.75·47-s − 0.280·51-s − 0.274·53-s − 0.269·55-s − 0.529·57-s − 1.56·59-s − 1.79·61-s − 1.48·65-s + 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 103488 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 103488 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.178293377\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.178293377\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−13.92134215703255, −13.20301403612485, −12.73704779427546, −12.48973420439216, −11.87120809665678, −11.16424767178777, −10.71627007260541, −10.12121488230078, −9.752672465878635, −9.328135152019063, −8.916805063147369, −8.194177867679492, −7.667054228918326, −7.378334393561165, −6.555323114449654, −6.210728452667866, −5.562706459164403, −4.921372628983167, −4.545335233182027, −3.822711548688057, −3.149624488797140, −2.420094770801016, −2.061141195947709, −1.611889945202637, −0.2920422198693138,
0.2920422198693138, 1.611889945202637, 2.061141195947709, 2.420094770801016, 3.149624488797140, 3.822711548688057, 4.545335233182027, 4.921372628983167, 5.562706459164403, 6.210728452667866, 6.555323114449654, 7.378334393561165, 7.667054228918326, 8.194177867679492, 8.916805063147369, 9.328135152019063, 9.752672465878635, 10.12121488230078, 10.71627007260541, 11.16424767178777, 11.87120809665678, 12.48973420439216, 12.73704779427546, 13.20301403612485, 13.92134215703255
