| L(s) = 1 | − 4·11-s + 2·13-s + 6·17-s + 19-s + 6·23-s + 2·29-s + 6·31-s − 10·37-s + 6·43-s + 6·47-s − 7·49-s − 10·53-s + 2·59-s − 6·61-s − 8·67-s − 12·71-s − 16·73-s + 14·79-s + 12·83-s − 4·89-s − 10·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 1.20·11-s + 0.554·13-s + 1.45·17-s + 0.229·19-s + 1.25·23-s + 0.371·29-s + 1.07·31-s − 1.64·37-s + 0.914·43-s + 0.875·47-s − 49-s − 1.37·53-s + 0.260·59-s − 0.768·61-s − 0.977·67-s − 1.42·71-s − 1.87·73-s + 1.57·79-s + 1.31·83-s − 0.423·89-s − 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 68400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 68400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 3 | \( 1 \) | |
| 5 | \( 1 \) | |
| 19 | \( 1 - T \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 - 2 T + p T^{2} \) | 1.59.ac |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 16 T + p T^{2} \) | 1.73.q |
| 79 | \( 1 - 14 T + p T^{2} \) | 1.79.ao |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 4 T + p T^{2} \) | 1.89.e |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−14.43141549534308, −13.69917003719094, −13.62049073206469, −12.84615684551720, −12.45567851421179, −11.98620425691060, −11.39064179714760, −10.79172138238073, −10.31831692756105, −10.09554956727816, −9.200853641017676, −8.891499967565878, −8.167442782952006, −7.714973326915497, −7.348599264543097, −6.580310954204445, −6.028509067134593, −5.427052893515706, −4.987982352386810, −4.430068329254750, −3.503399618074367, −3.072217327941363, −2.582596066758650, −1.540278850648151, −1.006371515369161, 0,
1.006371515369161, 1.540278850648151, 2.582596066758650, 3.072217327941363, 3.503399618074367, 4.430068329254750, 4.987982352386810, 5.427052893515706, 6.028509067134593, 6.580310954204445, 7.348599264543097, 7.714973326915497, 8.167442782952006, 8.891499967565878, 9.200853641017676, 10.09554956727816, 10.31831692756105, 10.79172138238073, 11.39064179714760, 11.98620425691060, 12.45567851421179, 12.84615684551720, 13.62049073206469, 13.69917003719094, 14.43141549534308
