| L(s) = 1 | + 3·3-s + 6·9-s + 11-s + 13-s − 3·17-s − 8·19-s + 4·23-s + 9·27-s − 3·29-s − 6·31-s + 3·33-s − 8·37-s + 3·39-s − 10·41-s + 12·43-s − 3·47-s − 9·51-s + 12·53-s − 24·57-s + 2·61-s − 4·67-s + 12·69-s + 12·71-s − 10·73-s + 13·79-s + 9·81-s − 9·87-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 2·9-s + 0.301·11-s + 0.277·13-s − 0.727·17-s − 1.83·19-s + 0.834·23-s + 1.73·27-s − 0.557·29-s − 1.07·31-s + 0.522·33-s − 1.31·37-s + 0.480·39-s − 1.56·41-s + 1.82·43-s − 0.437·47-s − 1.26·51-s + 1.64·53-s − 3.17·57-s + 0.256·61-s − 0.488·67-s + 1.44·69-s + 1.42·71-s − 1.17·73-s + 1.46·79-s + 81-s − 0.964·87-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
| \(L(1)\) |
\(\approx\) |
\(4.547495133\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.547495133\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - p T + p T^{2} \) | 1.3.ad |
| 11 | \( 1 - T + p T^{2} \) | 1.11.ab |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 3 T + p T^{2} \) | 1.29.d |
| 31 | \( 1 + 6 T + p T^{2} \) | 1.31.g |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 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.e |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 13 T + p T^{2} \) | 1.79.an |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 5 T + p T^{2} \) | 1.97.af |
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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
−14.01508318997658, −13.56120999891518, −13.12742136946203, −12.72297103388298, −12.29133338905016, −11.41930106935159, −10.95591755530336, −10.36392844847368, −10.01816161586744, −9.082440906116415, −8.905308588048368, −8.714493630660579, −7.992100752886506, −7.454699587094179, −6.907466377739160, −6.530770877781646, −5.728955742537122, −4.945121689814424, −4.341074109919075, −3.758084286571246, −3.450736165507659, −2.627944825017881, −2.013882841087742, −1.740813204327387, −0.5864898171656120,
0.5864898171656120, 1.740813204327387, 2.013882841087742, 2.627944825017881, 3.450736165507659, 3.758084286571246, 4.341074109919075, 4.945121689814424, 5.728955742537122, 6.530770877781646, 6.907466377739160, 7.454699587094179, 7.992100752886506, 8.714493630660579, 8.905308588048368, 9.082440906116415, 10.01816161586744, 10.36392844847368, 10.95591755530336, 11.41930106935159, 12.29133338905016, 12.72297103388298, 13.12742136946203, 13.56120999891518, 14.01508318997658
