| L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s − 7-s + 8-s + 9-s − 10-s + 12-s + 4·13-s − 14-s − 15-s + 16-s + 6·17-s + 18-s + 4·19-s − 20-s − 21-s − 6·23-s + 24-s + 25-s + 4·26-s + 27-s − 28-s + 9·29-s − 30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.288·12-s + 1.10·13-s − 0.267·14-s − 0.258·15-s + 1/4·16-s + 1.45·17-s + 0.235·18-s + 0.917·19-s − 0.223·20-s − 0.218·21-s − 1.25·23-s + 0.204·24-s + 1/5·25-s + 0.784·26-s + 0.192·27-s − 0.188·28-s + 1.67·29-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.264187806\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.264187806\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 3 | \( 1 - T \) | |
| 5 | \( 1 + T \) | |
| 7 | \( 1 + T \) | |
| 11 | \( 1 \) | |
| good | 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 9 T + p T^{2} \) | 1.29.aj |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + T + p T^{2} \) | 1.37.b |
| 41 | \( 1 + 3 T + p T^{2} \) | 1.41.d |
| 43 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - T + p T^{2} \) | 1.61.ab |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 - 3 T + p T^{2} \) | 1.71.ad |
| 73 | \( 1 - T + p T^{2} \) | 1.73.ab |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + 15 T + p T^{2} \) | 1.89.p |
| 97 | \( 1 - 11 T + p T^{2} \) | 1.97.al |
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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
−15.60567055218726, −14.63375700455209, −14.12648890058063, −13.87891720242019, −13.38657782399639, −12.60037871263635, −12.06424158942256, −11.94361699115805, −11.02444605301354, −10.49297342325992, −9.863680823171526, −9.470794898176633, −8.488256050214466, −8.113508705625934, −7.657086139693898, −6.896537507192516, −6.231648613458955, −5.825789486862003, −4.962075146331701, −4.323597869153568, −3.663822800209805, −3.162534777690422, −2.647285052866156, −1.503358936840389, −0.8464084496096321,
0.8464084496096321, 1.503358936840389, 2.647285052866156, 3.162534777690422, 3.663822800209805, 4.323597869153568, 4.962075146331701, 5.825789486862003, 6.231648613458955, 6.896537507192516, 7.657086139693898, 8.113508705625934, 8.488256050214466, 9.470794898176633, 9.863680823171526, 10.49297342325992, 11.02444605301354, 11.94361699115805, 12.06424158942256, 12.60037871263635, 13.38657782399639, 13.87891720242019, 14.12648890058063, 14.63375700455209, 15.60567055218726
