| L(s) = 1 | + 3·5-s + 3·11-s − 13-s + 3·17-s + 7·19-s + 9·23-s + 4·25-s + 3·29-s − 8·31-s − 37-s + 3·41-s + 43-s + 3·53-s + 9·55-s + 2·61-s − 3·65-s + 4·67-s − 12·71-s + 11·73-s + 16·79-s + 9·83-s + 9·85-s + 3·89-s + 21·95-s − 97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | + 1.34·5-s + 0.904·11-s − 0.277·13-s + 0.727·17-s + 1.60·19-s + 1.87·23-s + 4/5·25-s + 0.557·29-s − 1.43·31-s − 0.164·37-s + 0.468·41-s + 0.152·43-s + 0.412·53-s + 1.21·55-s + 0.256·61-s − 0.372·65-s + 0.488·67-s − 1.42·71-s + 1.28·73-s + 1.80·79-s + 0.987·83-s + 0.976·85-s + 0.317·89-s + 2.15·95-s − 0.101·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.809149092\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.809149092\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| good | 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 - 7 T + p T^{2} \) | 1.19.ah |
| 23 | \( 1 - 9 T + p T^{2} \) | 1.23.aj |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + T + p T^{2} \) | 1.37.b |
| 41 | \( 1 - 3 T + p T^{2} \) | 1.41.ad |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 3 T + p T^{2} \) | 1.53.ad |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 - 11 T + p T^{2} \) | 1.73.al |
| 79 | \( 1 - 16 T + p T^{2} \) | 1.79.aq |
| 83 | \( 1 - 9 T + p T^{2} \) | 1.83.aj |
| 89 | \( 1 - 3 T + p T^{2} \) | 1.89.ad |
| 97 | \( 1 + T + p T^{2} \) | 1.97.b |
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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
−14.23865942286583, −13.71553362136916, −13.40158176451429, −12.72776054092786, −12.33604354913618, −11.70522533575141, −11.24094552865418, −10.60184288713310, −10.13944054541166, −9.530737226959817, −9.128792342045044, −9.002755762644666, −7.931694761599788, −7.476476667121714, −6.824856224957609, −6.467103844253974, −5.681989748442133, −5.283170646410651, −4.933909318214639, −3.934122002261135, −3.328537499715393, −2.764043395208693, −2.009885023420562, −1.278996216804612, −0.8326078699526954,
0.8326078699526954, 1.278996216804612, 2.009885023420562, 2.764043395208693, 3.328537499715393, 3.934122002261135, 4.933909318214639, 5.283170646410651, 5.681989748442133, 6.467103844253974, 6.824856224957609, 7.476476667121714, 7.931694761599788, 9.002755762644666, 9.128792342045044, 9.530737226959817, 10.13944054541166, 10.60184288713310, 11.24094552865418, 11.70522533575141, 12.33604354913618, 12.72776054092786, 13.40158176451429, 13.71553362136916, 14.23865942286583
