| L(s) = 1 | + 3-s − 3·5-s + 9-s − 6·11-s − 3·15-s + 8·17-s − 19-s + 23-s + 4·25-s + 27-s − 5·29-s + 3·31-s − 6·33-s − 12·37-s + 10·41-s − 11·43-s − 3·45-s − 3·47-s + 8·51-s − 53-s + 18·55-s − 57-s − 8·59-s − 2·61-s − 2·67-s + 69-s − 8·71-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.34·5-s + 1/3·9-s − 1.80·11-s − 0.774·15-s + 1.94·17-s − 0.229·19-s + 0.208·23-s + 4/5·25-s + 0.192·27-s − 0.928·29-s + 0.538·31-s − 1.04·33-s − 1.97·37-s + 1.56·41-s − 1.67·43-s − 0.447·45-s − 0.437·47-s + 1.12·51-s − 0.137·53-s + 2.42·55-s − 0.132·57-s − 1.04·59-s − 0.256·61-s − 0.244·67-s + 0.120·69-s − 0.949·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 198744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 198744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7023085152\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7023085152\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 11 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 17 | \( 1 - 8 T + p T^{2} \) | 1.17.ai |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 - T + p T^{2} \) | 1.23.ab |
| 29 | \( 1 + 5 T + p T^{2} \) | 1.29.f |
| 31 | \( 1 - 3 T + p T^{2} \) | 1.31.ad |
| 37 | \( 1 + 12 T + p T^{2} \) | 1.37.m |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 + 11 T + p T^{2} \) | 1.43.l |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 + T + p T^{2} \) | 1.53.b |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 11 T + p T^{2} \) | 1.73.al |
| 79 | \( 1 - 13 T + p T^{2} \) | 1.79.an |
| 83 | \( 1 + 7 T + p T^{2} \) | 1.83.h |
| 89 | \( 1 - 7 T + p T^{2} \) | 1.89.ah |
| 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
−13.08480161871088, −12.46454116598339, −12.22453290870658, −11.82608329847170, −11.08899907530249, −10.70403003312443, −10.34962647000010, −9.746644819236559, −9.355521004747682, −8.525415629827870, −8.245914370500346, −7.759196163023854, −7.593428859686127, −7.071784941543520, −6.386230435122078, −5.572493062073273, −5.250260288990437, −4.731233192243826, −4.083699761877737, −3.391931475807767, −3.265580060102591, −2.626338968642782, −1.874835668734119, −1.137749246443271, −0.2407905218302523,
0.2407905218302523, 1.137749246443271, 1.874835668734119, 2.626338968642782, 3.265580060102591, 3.391931475807767, 4.083699761877737, 4.731233192243826, 5.250260288990437, 5.572493062073273, 6.386230435122078, 7.071784941543520, 7.593428859686127, 7.759196163023854, 8.245914370500346, 8.525415629827870, 9.355521004747682, 9.746644819236559, 10.34962647000010, 10.70403003312443, 11.08899907530249, 11.82608329847170, 12.22453290870658, 12.46454116598339, 13.08480161871088
