| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s + 2·7-s − 8-s − 2·9-s − 12-s − 4·13-s − 2·14-s + 16-s − 3·17-s + 2·18-s − 5·19-s − 2·21-s − 6·23-s + 24-s + 4·26-s + 5·27-s + 2·28-s + 2·31-s − 32-s + 3·34-s − 2·36-s − 2·37-s + 5·38-s + 4·39-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.755·7-s − 0.353·8-s − 2/3·9-s − 0.288·12-s − 1.10·13-s − 0.534·14-s + 1/4·16-s − 0.727·17-s + 0.471·18-s − 1.14·19-s − 0.436·21-s − 1.25·23-s + 0.204·24-s + 0.784·26-s + 0.962·27-s + 0.377·28-s + 0.359·31-s − 0.176·32-s + 0.514·34-s − 1/3·36-s − 0.328·37-s + 0.811·38-s + 0.640·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5385418801\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5385418801\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 11 | \( 1 \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 + 5 T + p T^{2} \) | 1.19.f |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 3 T + p T^{2} \) | 1.41.ad |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 13 T + p T^{2} \) | 1.67.an |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 11 T + p T^{2} \) | 1.73.al |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 + 9 T + p T^{2} \) | 1.83.j |
| 89 | \( 1 - 15 T + p T^{2} \) | 1.89.ap |
| 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
−8.220754621127534024773458290050, −7.51041179382499399787549934178, −6.49744316875353019324673001314, −6.24922641498919327876575296532, −5.09105535008620045303108360252, −4.74216535561610488800625171267, −3.59888450816132088527470084170, −2.41588854926150514505495479740, −1.87497066560588835960355678901, −0.41976510590696652595284766985,
0.41976510590696652595284766985, 1.87497066560588835960355678901, 2.41588854926150514505495479740, 3.59888450816132088527470084170, 4.74216535561610488800625171267, 5.09105535008620045303108360252, 6.24922641498919327876575296532, 6.49744316875353019324673001314, 7.51041179382499399787549934178, 8.220754621127534024773458290050
