| L(s) = 1 | + 3-s + 2·5-s + 9-s + 11-s + 2·15-s + 6·17-s − 4·23-s − 25-s + 27-s + 6·29-s + 4·31-s + 33-s − 2·37-s − 10·41-s + 4·43-s + 2·45-s + 8·47-s − 7·49-s + 6·51-s − 6·53-s + 2·55-s + 4·59-s − 2·61-s − 4·67-s − 4·69-s − 8·71-s − 14·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894·5-s + 1/3·9-s + 0.301·11-s + 0.516·15-s + 1.45·17-s − 0.834·23-s − 1/5·25-s + 0.192·27-s + 1.11·29-s + 0.718·31-s + 0.174·33-s − 0.328·37-s − 1.56·41-s + 0.609·43-s + 0.298·45-s + 1.16·47-s − 49-s + 0.840·51-s − 0.824·53-s + 0.269·55-s + 0.520·59-s − 0.256·61-s − 0.488·67-s − 0.481·69-s − 0.949·71-s − 1.63·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 89232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 89232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.461561970\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.461561970\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 11 | \( 1 - T \) | |
| 13 | \( 1 \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−13.89490502360512, −13.56443576315842, −12.98638475605629, −12.37546712286008, −11.95714887618450, −11.60943589808210, −10.67088838800723, −10.23078621260426, −9.901922523475486, −9.538981824217764, −8.735283611106581, −8.530493229674074, −7.738378928405270, −7.465445019064224, −6.669494975532601, −6.108323206801197, −5.814541606060342, −5.014269270884978, −4.566833314993640, −3.770808746756883, −3.240944700792827, −2.693780339141857, −1.913605972919320, −1.472572751965747, −0.6520318626826485,
0.6520318626826485, 1.472572751965747, 1.913605972919320, 2.693780339141857, 3.240944700792827, 3.770808746756883, 4.566833314993640, 5.014269270884978, 5.814541606060342, 6.108323206801197, 6.669494975532601, 7.465445019064224, 7.738378928405270, 8.530493229674074, 8.735283611106581, 9.538981824217764, 9.901922523475486, 10.23078621260426, 10.67088838800723, 11.60943589808210, 11.95714887618450, 12.37546712286008, 12.98638475605629, 13.56443576315842, 13.89490502360512
