| L(s) = 1 | + 3-s + 5-s + 3·7-s − 2·9-s − 5·11-s + 15-s + 5·17-s + 19-s + 3·21-s + 23-s + 25-s − 5·27-s − 9·29-s + 4·31-s − 5·33-s + 3·35-s − 3·37-s − 41-s + 3·43-s − 2·45-s + 8·47-s + 2·49-s + 5·51-s + 10·53-s − 5·55-s + 57-s − 3·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1.13·7-s − 2/3·9-s − 1.50·11-s + 0.258·15-s + 1.21·17-s + 0.229·19-s + 0.654·21-s + 0.208·23-s + 1/5·25-s − 0.962·27-s − 1.67·29-s + 0.718·31-s − 0.870·33-s + 0.507·35-s − 0.493·37-s − 0.156·41-s + 0.457·43-s − 0.298·45-s + 1.16·47-s + 2/7·49-s + 0.700·51-s + 1.37·53-s − 0.674·55-s + 0.132·57-s − 0.390·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.900527963\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.900527963\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 13 | \( 1 \) | |
| good | 3 | \( 1 - T + p T^{2} \) | 1.3.ab |
| 7 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 11 | \( 1 + 5 T + p T^{2} \) | 1.11.f |
| 17 | \( 1 - 5 T + p T^{2} \) | 1.17.af |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 - T + p T^{2} \) | 1.23.ab |
| 29 | \( 1 + 9 T + p T^{2} \) | 1.29.j |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 3 T + p T^{2} \) | 1.37.d |
| 41 | \( 1 + T + p T^{2} \) | 1.41.b |
| 43 | \( 1 - 3 T + p T^{2} \) | 1.43.ad |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 - 7 T + p T^{2} \) | 1.61.ah |
| 67 | \( 1 - 9 T + p T^{2} \) | 1.67.aj |
| 71 | \( 1 + 7 T + p T^{2} \) | 1.71.h |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 - 16 T + p T^{2} \) | 1.79.aq |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 15 T + p T^{2} \) | 1.89.ap |
| 97 | \( 1 + 7 T + p T^{2} \) | 1.97.h |
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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
−16.19996811960005, −15.49209902649794, −14.90872592555520, −14.62234896527861, −13.82967186738561, −13.66318885437048, −12.92005485870577, −12.25562837435569, −11.64044329719434, −10.96111381241260, −10.57804502636371, −9.832120452302555, −9.269290720929076, −8.520197792313890, −8.020433891779465, −7.663389185229541, −6.955911321723832, −5.709904201371578, −5.520878182317374, −4.951647814955785, −3.938873525648673, −3.157167455993323, −2.439913231780743, −1.863875934230161, −0.7241633396757935,
0.7241633396757935, 1.863875934230161, 2.439913231780743, 3.157167455993323, 3.938873525648673, 4.951647814955785, 5.520878182317374, 5.709904201371578, 6.955911321723832, 7.663389185229541, 8.020433891779465, 8.520197792313890, 9.269290720929076, 9.832120452302555, 10.57804502636371, 10.96111381241260, 11.64044329719434, 12.25562837435569, 12.92005485870577, 13.66318885437048, 13.82967186738561, 14.62234896527861, 14.90872592555520, 15.49209902649794, 16.19996811960005
