| L(s) = 1 | + 2·3-s − 7-s + 9-s − 4·11-s − 4·13-s + 2·17-s − 6·19-s − 2·21-s + 8·23-s − 4·27-s − 2·29-s + 4·31-s − 8·33-s + 10·37-s − 8·39-s − 10·41-s − 4·43-s + 4·47-s + 49-s + 4·51-s − 2·53-s − 12·57-s + 10·59-s + 8·61-s − 63-s + 8·67-s + 16·69-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.377·7-s + 1/3·9-s − 1.20·11-s − 1.10·13-s + 0.485·17-s − 1.37·19-s − 0.436·21-s + 1.66·23-s − 0.769·27-s − 0.371·29-s + 0.718·31-s − 1.39·33-s + 1.64·37-s − 1.28·39-s − 1.56·41-s − 0.609·43-s + 0.583·47-s + 1/7·49-s + 0.560·51-s − 0.274·53-s − 1.58·57-s + 1.30·59-s + 1.02·61-s − 0.125·63-s + 0.977·67-s + 1.92·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.172508471\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.172508471\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 - 10 T + p T^{2} \) | 1.59.ak |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 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 | \( 1 - 16 T + p T^{2} \) | 1.79.aq |
| 83 | \( 1 + 2 T + p T^{2} \) | 1.83.c |
| 89 | \( 1 - 18 T + p T^{2} \) | 1.89.as |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−16.61629759078180, −15.64120137768145, −15.21866389885818, −14.76068946915782, −14.40648546115125, −13.44934923456218, −13.15904937149064, −12.73701024172080, −11.95714420445907, −11.22054234416087, −10.51319172944293, −9.972798293742680, −9.427398691232698, −8.784907492027574, −8.145063543883725, −7.766682371350217, −6.999698319776016, −6.395966798342721, −5.332266015230958, −4.924205237564143, −3.955060542115768, −3.177977175359349, −2.564571396737269, −2.102960445424548, −0.6030981418250249,
0.6030981418250249, 2.102960445424548, 2.564571396737269, 3.177977175359349, 3.955060542115768, 4.924205237564143, 5.332266015230958, 6.395966798342721, 6.999698319776016, 7.766682371350217, 8.145063543883725, 8.784907492027574, 9.427398691232698, 9.972798293742680, 10.51319172944293, 11.22054234416087, 11.95714420445907, 12.73701024172080, 13.15904937149064, 13.44934923456218, 14.40648546115125, 14.76068946915782, 15.21866389885818, 15.64120137768145, 16.61629759078180
