| L(s) = 1 | + 2·3-s + 5-s + 9-s + 4·11-s + 2·13-s + 2·15-s − 6·17-s + 4·19-s + 25-s − 4·27-s − 4·29-s − 8·31-s + 8·33-s + 8·37-s + 4·39-s + 2·41-s − 2·43-s + 45-s − 8·47-s − 12·51-s + 12·53-s + 4·55-s + 8·57-s − 10·59-s − 8·61-s + 2·65-s − 8·67-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.447·5-s + 1/3·9-s + 1.20·11-s + 0.554·13-s + 0.516·15-s − 1.45·17-s + 0.917·19-s + 1/5·25-s − 0.769·27-s − 0.742·29-s − 1.43·31-s + 1.39·33-s + 1.31·37-s + 0.640·39-s + 0.312·41-s − 0.304·43-s + 0.149·45-s − 1.16·47-s − 1.68·51-s + 1.64·53-s + 0.539·55-s + 1.05·57-s − 1.30·59-s − 1.02·61-s + 0.248·65-s − 0.977·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 309680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 309680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.169607930\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.169607930\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| 79 | \( 1 - T \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 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 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 83 | \( 1 + 8 T + p T^{2} \) | 1.83.i |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−12.90536805361826, −12.26505257864536, −11.55594511434273, −11.38073110423000, −10.89436788415271, −10.34324390846535, −9.574413072301323, −9.262910325708734, −9.197848389849659, −8.596616908905775, −8.123617380506485, −7.640076310622030, −6.989823015352515, −6.781173969078080, −5.908922981746075, −5.843969271131617, −5.020495250787045, −4.332740319107082, −3.972046101230072, −3.455956440231368, −2.917915275243852, −2.401491480348277, −1.663782501055352, −1.495904165498593, −0.4664672845169670,
0.4664672845169670, 1.495904165498593, 1.663782501055352, 2.401491480348277, 2.917915275243852, 3.455956440231368, 3.972046101230072, 4.332740319107082, 5.020495250787045, 5.843969271131617, 5.908922981746075, 6.781173969078080, 6.989823015352515, 7.640076310622030, 8.123617380506485, 8.596616908905775, 9.197848389849659, 9.262910325708734, 9.574413072301323, 10.34324390846535, 10.89436788415271, 11.38073110423000, 11.55594511434273, 12.26505257864536, 12.90536805361826
