| L(s) = 1 | − 2·7-s + 4·11-s + 6·13-s + 4·17-s − 2·19-s + 23-s − 2·29-s − 4·31-s − 2·37-s − 2·41-s + 10·43-s − 3·49-s − 12·53-s − 12·59-s − 6·61-s − 10·67-s + 8·71-s + 14·73-s − 8·77-s − 10·79-s − 12·83-s + 16·89-s − 12·91-s + 10·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 0.755·7-s + 1.20·11-s + 1.66·13-s + 0.970·17-s − 0.458·19-s + 0.208·23-s − 0.371·29-s − 0.718·31-s − 0.328·37-s − 0.312·41-s + 1.52·43-s − 3/7·49-s − 1.64·53-s − 1.56·59-s − 0.768·61-s − 1.22·67-s + 0.949·71-s + 1.63·73-s − 0.911·77-s − 1.12·79-s − 1.31·83-s + 1.69·89-s − 1.25·91-s + 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 82800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 82800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.484765517\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.484765517\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 23 | \( 1 - T \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 + 10 T + p T^{2} \) | 1.67.k |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 16 T + p T^{2} \) | 1.89.aq |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−14.10911911363619, −13.49608447034733, −12.81904642562793, −12.61519319123556, −12.02400714708150, −11.44317082417379, −10.89460304577190, −10.65569325753774, −9.840124271347609, −9.363768433391210, −9.021502158409600, −8.519122289746752, −7.814432677734735, −7.384504814037178, −6.531380693806991, −6.319650059893136, −5.866999361154855, −5.191260428801903, −4.349040692536010, −3.857683205346403, −3.360390907547747, −2.924197275998339, −1.748419081187292, −1.390076015865048, −0.5318484666035818,
0.5318484666035818, 1.390076015865048, 1.748419081187292, 2.924197275998339, 3.360390907547747, 3.857683205346403, 4.349040692536010, 5.191260428801903, 5.866999361154855, 6.319650059893136, 6.531380693806991, 7.384504814037178, 7.814432677734735, 8.519122289746752, 9.021502158409600, 9.363768433391210, 9.840124271347609, 10.65569325753774, 10.89460304577190, 11.44317082417379, 12.02400714708150, 12.61519319123556, 12.81904642562793, 13.49608447034733, 14.10911911363619
