| L(s) = 1 | + 3-s + 4·5-s + 9-s + 11-s + 4·15-s + 6·17-s + 4·19-s − 6·23-s + 11·25-s + 27-s − 6·29-s + 33-s − 6·37-s + 10·41-s + 8·43-s + 4·45-s − 6·47-s + 6·51-s + 12·53-s + 4·55-s + 4·57-s − 8·59-s + 4·61-s + 12·67-s − 6·69-s + 10·71-s − 2·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.78·5-s + 1/3·9-s + 0.301·11-s + 1.03·15-s + 1.45·17-s + 0.917·19-s − 1.25·23-s + 11/5·25-s + 0.192·27-s − 1.11·29-s + 0.174·33-s − 0.986·37-s + 1.56·41-s + 1.21·43-s + 0.596·45-s − 0.875·47-s + 0.840·51-s + 1.64·53-s + 0.539·55-s + 0.529·57-s − 1.04·59-s + 0.512·61-s + 1.46·67-s − 0.722·69-s + 1.18·71-s − 0.234·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 103488 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 103488 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.478462413\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.478462413\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| good | 5 | \( 1 - 4 T + p T^{2} \) | 1.5.ae |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 - 4 T + p T^{2} \) | 1.61.ae |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 - 10 T + p T^{2} \) | 1.71.ak |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 2 T + p T^{2} \) | 1.79.ac |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−13.73261574064791, −13.45747570143135, −12.81935813182426, −12.23785159461657, −12.07663761433619, −11.04869133398629, −10.75799502593367, −9.983444108836390, −9.764305461427147, −9.405042780725600, −8.974521431429247, −8.205753100915277, −7.795846354405414, −7.173534519158171, −6.657311467204983, −5.957205019203249, −5.562478675927096, −5.307782559261981, −4.408715737665648, −3.693653799142194, −3.240765371977768, −2.423523082204230, −2.066287885553603, −1.370876068609244, −0.7970849812773346,
0.7970849812773346, 1.370876068609244, 2.066287885553603, 2.423523082204230, 3.240765371977768, 3.693653799142194, 4.408715737665648, 5.307782559261981, 5.562478675927096, 5.957205019203249, 6.657311467204983, 7.173534519158171, 7.795846354405414, 8.205753100915277, 8.974521431429247, 9.405042780725600, 9.764305461427147, 9.983444108836390, 10.75799502593367, 11.04869133398629, 12.07663761433619, 12.23785159461657, 12.81935813182426, 13.45747570143135, 13.73261574064791
