| L(s) = 1 | − 3-s − 2·5-s + 9-s + 4·11-s + 4·13-s + 2·15-s + 2·17-s − 4·23-s − 25-s − 27-s + 4·31-s − 4·33-s − 2·37-s − 4·39-s + 41-s − 12·43-s − 2·45-s − 2·47-s − 2·51-s + 4·53-s − 8·55-s + 4·59-s + 10·61-s − 8·65-s − 8·67-s + 4·69-s + 10·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s + 1/3·9-s + 1.20·11-s + 1.10·13-s + 0.516·15-s + 0.485·17-s − 0.834·23-s − 1/5·25-s − 0.192·27-s + 0.718·31-s − 0.696·33-s − 0.328·37-s − 0.640·39-s + 0.156·41-s − 1.82·43-s − 0.298·45-s − 0.291·47-s − 0.280·51-s + 0.549·53-s − 1.07·55-s + 0.520·59-s + 1.28·61-s − 0.992·65-s − 0.977·67-s + 0.481·69-s + 1.18·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 385728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 385728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.457911992\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.457911992\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| 41 | \( 1 - T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 - 4 T + p T^{2} \) | 1.53.ae |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 10 T + p T^{2} \) | 1.71.ak |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - 14 T + p T^{2} \) | 1.79.ao |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 - 18 T + p T^{2} \) | 1.97.as |
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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
−12.37238286963913, −11.81644823451104, −11.56204019348160, −11.40639406717744, −10.73103177383611, −10.16034930232394, −9.927916720985716, −9.303126620891879, −8.653042365392257, −8.481530013604293, −7.835011874603382, −7.511910089627263, −6.802374316769187, −6.434136980757652, −6.136505658774469, −5.499947293067797, −4.951913604910610, −4.423045230180176, −3.832129796366950, −3.618150897491593, −3.152525761014849, −2.068138362966569, −1.707945651791584, −0.8518034122381208, −0.5573198316916465,
0.5573198316916465, 0.8518034122381208, 1.707945651791584, 2.068138362966569, 3.152525761014849, 3.618150897491593, 3.832129796366950, 4.423045230180176, 4.951913604910610, 5.499947293067797, 6.136505658774469, 6.434136980757652, 6.802374316769187, 7.511910089627263, 7.835011874603382, 8.481530013604293, 8.653042365392257, 9.303126620891879, 9.927916720985716, 10.16034930232394, 10.73103177383611, 11.40639406717744, 11.56204019348160, 11.81644823451104, 12.37238286963913
