| L(s) = 1 | − 2·3-s + 9-s − 4·11-s − 6·13-s − 4·17-s + 6·19-s + 4·23-s + 4·27-s + 6·29-s + 4·31-s + 8·33-s − 6·37-s + 12·39-s − 4·41-s + 12·43-s − 12·47-s + 8·51-s + 6·53-s − 12·57-s + 6·59-s − 6·61-s + 12·67-s − 8·69-s + 8·71-s − 11·81-s − 6·83-s − 12·87-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1/3·9-s − 1.20·11-s − 1.66·13-s − 0.970·17-s + 1.37·19-s + 0.834·23-s + 0.769·27-s + 1.11·29-s + 0.718·31-s + 1.39·33-s − 0.986·37-s + 1.92·39-s − 0.624·41-s + 1.82·43-s − 1.75·47-s + 1.12·51-s + 0.824·53-s − 1.58·57-s + 0.781·59-s − 0.768·61-s + 1.46·67-s − 0.963·69-s + 0.949·71-s − 1.22·81-s − 0.658·83-s − 1.28·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7828625770\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7828625770\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + 4 T + p T^{2} \) | 1.41.e |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + p T^{2} \) | 1.73.a |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 - 16 T + p T^{2} \) | 1.89.aq |
| 97 | \( 1 + 12 T + p T^{2} \) | 1.97.m |
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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
−14.12629351490102, −13.41502785019472, −12.90258742244383, −12.45617737305412, −11.95672679913964, −11.58945032759386, −11.08914495084366, −10.41011073647992, −10.27495656739589, −9.571014719039612, −9.103434075639346, −8.295083860976212, −7.896595859143263, −7.159409019408548, −6.860211710859883, −6.305972931149132, −5.448677959120764, −5.141358176831669, −4.888633544462919, −4.204128371371293, −3.132103076668880, −2.721134881624673, −2.111951462203254, −0.9996517269719899, −0.3671699377371634,
0.3671699377371634, 0.9996517269719899, 2.111951462203254, 2.721134881624673, 3.132103076668880, 4.204128371371293, 4.888633544462919, 5.141358176831669, 5.448677959120764, 6.305972931149132, 6.860211710859883, 7.159409019408548, 7.896595859143263, 8.295083860976212, 9.103434075639346, 9.571014719039612, 10.27495656739589, 10.41011073647992, 11.08914495084366, 11.58945032759386, 11.95672679913964, 12.45617737305412, 12.90258742244383, 13.41502785019472, 14.12629351490102
