| L(s) = 1 | − 2·3-s + 2·5-s − 4·7-s + 9-s − 6·11-s − 6·13-s − 4·15-s − 6·17-s − 2·19-s + 8·21-s + 4·23-s − 25-s + 4·27-s − 4·31-s + 12·33-s − 8·35-s − 10·37-s + 12·39-s − 10·41-s + 2·43-s + 2·45-s − 4·47-s + 9·49-s + 12·51-s + 2·53-s − 12·55-s + 4·57-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.894·5-s − 1.51·7-s + 1/3·9-s − 1.80·11-s − 1.66·13-s − 1.03·15-s − 1.45·17-s − 0.458·19-s + 1.74·21-s + 0.834·23-s − 1/5·25-s + 0.769·27-s − 0.718·31-s + 2.08·33-s − 1.35·35-s − 1.64·37-s + 1.92·39-s − 1.56·41-s + 0.304·43-s + 0.298·45-s − 0.583·47-s + 9/7·49-s + 1.68·51-s + 0.274·53-s − 1.61·55-s + 0.529·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 107648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 107648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 29 | \( 1 \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + 14 T + p T^{2} \) | 1.59.o |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
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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
−13.61386348335062, −13.34489880797313, −12.95322868916883, −12.44411390328028, −12.17481891290921, −11.47766202973168, −10.77374283435174, −10.49307603681975, −10.14187555450340, −9.700357626472621, −8.971792079821882, −8.787138145615456, −7.723495119386206, −7.274361184268632, −6.745554716731963, −6.331388888918403, −5.840392122848820, −5.253748427330277, −4.939271654071723, −4.456633738650668, −3.278207672791350, −2.924546541719043, −2.274473617625441, −1.756462731374444, −0.3011351592497334, 0,
0.3011351592497334, 1.756462731374444, 2.274473617625441, 2.924546541719043, 3.278207672791350, 4.456633738650668, 4.939271654071723, 5.253748427330277, 5.840392122848820, 6.331388888918403, 6.745554716731963, 7.274361184268632, 7.723495119386206, 8.787138145615456, 8.971792079821882, 9.700357626472621, 10.14187555450340, 10.49307603681975, 10.77374283435174, 11.47766202973168, 12.17481891290921, 12.44411390328028, 12.95322868916883, 13.34489880797313, 13.61386348335062
