| L(s) = 1 | + 3-s + 2·5-s + 4·7-s + 9-s − 11-s + 2·15-s + 4·17-s − 6·19-s + 4·21-s − 6·23-s − 25-s + 27-s + 6·29-s + 6·31-s − 33-s + 8·35-s + 4·37-s + 10·41-s + 8·43-s + 2·45-s − 12·47-s + 9·49-s + 4·51-s − 12·53-s − 2·55-s − 6·57-s + 12·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894·5-s + 1.51·7-s + 1/3·9-s − 0.301·11-s + 0.516·15-s + 0.970·17-s − 1.37·19-s + 0.872·21-s − 1.25·23-s − 1/5·25-s + 0.192·27-s + 1.11·29-s + 1.07·31-s − 0.174·33-s + 1.35·35-s + 0.657·37-s + 1.56·41-s + 1.21·43-s + 0.298·45-s − 1.75·47-s + 9/7·49-s + 0.560·51-s − 1.64·53-s − 0.269·55-s − 0.794·57-s + 1.56·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 178464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 178464 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.066605245\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.066605245\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 8 T + p T^{2} \) | 1.73.ai |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−13.17911713243066, −12.66444027315618, −12.45779261005805, −11.54967250688140, −11.41509139093903, −10.74163437830556, −10.23935925797412, −9.816385836421773, −9.555048333723021, −8.682597101693589, −8.307505858070900, −7.999599230948102, −7.662747170422241, −6.838699566499844, −6.270991685421816, −5.881012574374886, −5.276050782567290, −4.730682822445418, −4.273795980166190, −3.780449730269703, −2.868562365572377, −2.294301527292331, −2.028788375151743, −1.310523742671377, −0.6921402763927161,
0.6921402763927161, 1.310523742671377, 2.028788375151743, 2.294301527292331, 2.868562365572377, 3.780449730269703, 4.273795980166190, 4.730682822445418, 5.276050782567290, 5.881012574374886, 6.270991685421816, 6.838699566499844, 7.662747170422241, 7.999599230948102, 8.307505858070900, 8.682597101693589, 9.555048333723021, 9.816385836421773, 10.23935925797412, 10.74163437830556, 11.41509139093903, 11.54967250688140, 12.45779261005805, 12.66444027315618, 13.17911713243066
