| L(s) = 1 | + 3-s + 4·5-s − 2·7-s + 9-s − 11-s + 2·13-s + 4·15-s − 6·17-s + 4·19-s − 2·21-s + 11·25-s + 27-s − 2·29-s + 6·31-s − 33-s − 8·35-s − 37-s + 2·39-s + 12·41-s + 4·43-s + 4·45-s − 3·49-s − 6·51-s + 6·53-s − 4·55-s + 4·57-s − 12·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.78·5-s − 0.755·7-s + 1/3·9-s − 0.301·11-s + 0.554·13-s + 1.03·15-s − 1.45·17-s + 0.917·19-s − 0.436·21-s + 11/5·25-s + 0.192·27-s − 0.371·29-s + 1.07·31-s − 0.174·33-s − 1.35·35-s − 0.164·37-s + 0.320·39-s + 1.87·41-s + 0.609·43-s + 0.596·45-s − 3/7·49-s − 0.840·51-s + 0.824·53-s − 0.539·55-s + 0.529·57-s − 1.56·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 78144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.642346524\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.642346524\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 37 | \( 1 + T \) | |
| good | 5 | \( 1 - 4 T + p T^{2} \) | 1.5.ae |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 41 | \( 1 - 12 T + p T^{2} \) | 1.41.am |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 8 T + p T^{2} \) | 1.89.i |
| 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
−13.94157993982707, −13.53166668902262, −13.23678667082667, −12.64677725465409, −12.36774517116865, −11.30647734901418, −10.98989090706626, −10.34066516934488, −9.923120510469362, −9.459481430449891, −8.999948981308346, −8.789790865324175, −7.862750739293127, −7.385115164883095, −6.595516770891792, −6.290394715561323, −5.884625147501925, −5.165885143695375, −4.622610918803718, −3.900017988353652, −3.106560495933308, −2.630734238254006, −2.154600838050037, −1.442380147802556, −0.6742153808903359,
0.6742153808903359, 1.442380147802556, 2.154600838050037, 2.630734238254006, 3.106560495933308, 3.900017988353652, 4.622610918803718, 5.165885143695375, 5.884625147501925, 6.290394715561323, 6.595516770891792, 7.385115164883095, 7.862750739293127, 8.789790865324175, 8.999948981308346, 9.459481430449891, 9.923120510469362, 10.34066516934488, 10.98989090706626, 11.30647734901418, 12.36774517116865, 12.64677725465409, 13.23678667082667, 13.53166668902262, 13.94157993982707
