| L(s) = 1 | + 4·7-s − 6·11-s + 13-s − 2·17-s + 19-s − 5·25-s − 2·29-s + 8·31-s + 2·37-s + 4·41-s + 4·43-s − 6·47-s + 9·49-s + 2·53-s + 10·59-s − 10·61-s − 8·67-s − 2·71-s + 14·73-s − 24·77-s − 8·79-s + 10·83-s + 4·91-s + 6·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | + 1.51·7-s − 1.80·11-s + 0.277·13-s − 0.485·17-s + 0.229·19-s − 25-s − 0.371·29-s + 1.43·31-s + 0.328·37-s + 0.624·41-s + 0.609·43-s − 0.875·47-s + 9/7·49-s + 0.274·53-s + 1.30·59-s − 1.28·61-s − 0.977·67-s − 0.237·71-s + 1.63·73-s − 2.73·77-s − 0.900·79-s + 1.09·83-s + 0.419·91-s + 0.609·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 142272 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 142272 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.260571792\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.260571792\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| 19 | \( 1 - T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 4 T + p T^{2} \) | 1.41.ae |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 - 10 T + p T^{2} \) | 1.59.ak |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + 2 T + p T^{2} \) | 1.71.c |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 10 T + p T^{2} \) | 1.83.ak |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
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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
−13.25179756948515, −13.18869554671148, −12.37058566300569, −11.90411008738906, −11.43971552421868, −10.88303325109886, −10.75742275689114, −10.03006722992356, −9.666047622389646, −8.914164212143673, −8.343107926025306, −8.073669734742265, −7.601987106846874, −7.267025935652394, −6.358314919330267, −5.892087962791319, −5.293744150284997, −4.918486149672538, −4.426374033132884, −3.866310841204152, −2.995112927267930, −2.452530047061752, −1.985464514822471, −1.251210860023404, −0.4541791675433418,
0.4541791675433418, 1.251210860023404, 1.985464514822471, 2.452530047061752, 2.995112927267930, 3.866310841204152, 4.426374033132884, 4.918486149672538, 5.293744150284997, 5.892087962791319, 6.358314919330267, 7.267025935652394, 7.601987106846874, 8.073669734742265, 8.343107926025306, 8.914164212143673, 9.666047622389646, 10.03006722992356, 10.75742275689114, 10.88303325109886, 11.43971552421868, 11.90411008738906, 12.37058566300569, 13.18869554671148, 13.25179756948515
