| L(s) = 1 | − 2-s + 4-s + 4·5-s − 8-s − 4·10-s + 11-s − 13-s + 16-s − 3·17-s − 19-s + 4·20-s − 22-s − 6·23-s + 11·25-s + 26-s + 9·29-s − 8·31-s − 32-s + 3·34-s − 8·37-s + 38-s − 4·40-s + 10·43-s + 44-s + 6·46-s + 11·47-s − 11·50-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 1.78·5-s − 0.353·8-s − 1.26·10-s + 0.301·11-s − 0.277·13-s + 1/4·16-s − 0.727·17-s − 0.229·19-s + 0.894·20-s − 0.213·22-s − 1.25·23-s + 11/5·25-s + 0.196·26-s + 1.67·29-s − 1.43·31-s − 0.176·32-s + 0.514·34-s − 1.31·37-s + 0.162·38-s − 0.632·40-s + 1.52·43-s + 0.150·44-s + 0.884·46-s + 1.60·47-s − 1.55·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11466 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11466 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.113324451\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.113324451\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 5 | \( 1 - 4 T + p T^{2} \) | 1.5.ae |
| 11 | \( 1 - T + p T^{2} \) | 1.11.ab |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 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.i |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 47 | \( 1 - 11 T + p T^{2} \) | 1.47.al |
| 53 | \( 1 + T + p T^{2} \) | 1.53.b |
| 59 | \( 1 - 5 T + p T^{2} \) | 1.59.af |
| 61 | \( 1 + 15 T + p T^{2} \) | 1.61.p |
| 67 | \( 1 + 5 T + p T^{2} \) | 1.67.f |
| 71 | \( 1 - 15 T + p T^{2} \) | 1.71.ap |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + 2 T + p T^{2} \) | 1.79.c |
| 83 | \( 1 - 8 T + p T^{2} \) | 1.83.ai |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−16.78313808824781, −15.82386048234073, −15.58278682494852, −14.52733631591386, −14.15900546359485, −13.75067895581244, −13.02956155498210, −12.36404924389414, −11.95412998447037, −10.84910590055673, −10.57149922876382, −10.04855843306735, −9.278736121091707, −9.074061540422746, −8.383959202078975, −7.488527418182214, −6.822315521823672, −6.217437256198945, −5.772217959788336, −5.008802340313426, −4.146682042175874, −3.052006356168045, −2.160199982548933, −1.865631483144967, −0.7369011465816611,
0.7369011465816611, 1.865631483144967, 2.160199982548933, 3.052006356168045, 4.146682042175874, 5.008802340313426, 5.772217959788336, 6.217437256198945, 6.822315521823672, 7.488527418182214, 8.383959202078975, 9.074061540422746, 9.278736121091707, 10.04855843306735, 10.57149922876382, 10.84910590055673, 11.95412998447037, 12.36404924389414, 13.02956155498210, 13.75067895581244, 14.15900546359485, 14.52733631591386, 15.58278682494852, 15.82386048234073, 16.78313808824781
