| L(s) = 1 | − 2-s − 3-s + 4-s + 2·5-s + 6-s + 7-s − 8-s + 9-s − 2·10-s − 6·11-s − 12-s + 6·13-s − 14-s − 2·15-s + 16-s − 18-s − 6·19-s + 2·20-s − 21-s + 6·22-s + 6·23-s + 24-s − 25-s − 6·26-s − 27-s + 28-s − 6·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.894·5-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.632·10-s − 1.80·11-s − 0.288·12-s + 1.66·13-s − 0.267·14-s − 0.516·15-s + 1/4·16-s − 0.235·18-s − 1.37·19-s + 0.447·20-s − 0.218·21-s + 1.27·22-s + 1.25·23-s + 0.204·24-s − 1/5·25-s − 1.17·26-s − 0.192·27-s + 0.188·28-s − 1.11·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12138 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.251465574\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.251465574\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 7 | \( 1 - T \) | |
| 17 | \( 1 \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 11 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 - 2 T + p T^{2} \) | 1.71.ac |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 4 T + p T^{2} \) | 1.89.ae |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
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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
−16.33841961849884, −15.92907981825175, −15.36161854971024, −14.93623395916552, −13.91514616264390, −13.40500110090545, −12.99927505283689, −12.46257199188280, −11.45775970084466, −11.00403151875648, −10.60144212996990, −10.14719137385119, −9.453632793523084, −8.567181470093894, −8.358028288834180, −7.563382016851226, −6.724833575645874, −6.251896762624331, −5.492024808294301, −5.153251221419902, −4.128224016938423, −3.140451022927734, −2.247247372006080, −1.621905321313937, −0.5879888441093972,
0.5879888441093972, 1.621905321313937, 2.247247372006080, 3.140451022927734, 4.128224016938423, 5.153251221419902, 5.492024808294301, 6.251896762624331, 6.724833575645874, 7.563382016851226, 8.358028288834180, 8.567181470093894, 9.453632793523084, 10.14719137385119, 10.60144212996990, 11.00403151875648, 11.45775970084466, 12.46257199188280, 12.99927505283689, 13.40500110090545, 13.91514616264390, 14.93623395916552, 15.36161854971024, 15.92907981825175, 16.33841961849884
