| L(s) = 1 | − 2·5-s + 2·7-s − 3·9-s − 4·17-s − 3·19-s + 9·23-s − 25-s + 9·29-s + 4·31-s − 4·35-s − 6·37-s − 6·41-s + 11·43-s + 6·45-s − 3·47-s − 3·49-s + 14·53-s + 12·59-s + 2·61-s − 6·63-s + 2·67-s − 13·71-s − 10·73-s + 8·79-s + 9·81-s + 7·83-s + 8·85-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.755·7-s − 9-s − 0.970·17-s − 0.688·19-s + 1.87·23-s − 1/5·25-s + 1.67·29-s + 0.718·31-s − 0.676·35-s − 0.986·37-s − 0.937·41-s + 1.67·43-s + 0.894·45-s − 0.437·47-s − 3/7·49-s + 1.92·53-s + 1.56·59-s + 0.256·61-s − 0.755·63-s + 0.244·67-s − 1.54·71-s − 1.17·73-s + 0.900·79-s + 81-s + 0.768·83-s + 0.867·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 327184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 327184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.009575336\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.009575336\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 11 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 + 3 T + p T^{2} \) | 1.19.d |
| 23 | \( 1 - 9 T + p T^{2} \) | 1.23.aj |
| 29 | \( 1 - 9 T + p T^{2} \) | 1.29.aj |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 11 T + p T^{2} \) | 1.43.al |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 - 14 T + p T^{2} \) | 1.53.ao |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 + 13 T + p T^{2} \) | 1.71.n |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 7 T + p T^{2} \) | 1.83.ah |
| 89 | \( 1 + 3 T + p T^{2} \) | 1.89.d |
| 97 | \( 1 + 7 T + p T^{2} \) | 1.97.h |
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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
−12.53587588026727, −11.99107481001686, −11.57360833379216, −11.45728680456608, −10.75604438376466, −10.57248010552454, −9.994869466876843, −9.185307555141827, −8.733361202292963, −8.529051396927011, −8.163857532378593, −7.559508322670770, −6.955469636222379, −6.725592016323210, −6.080423456098427, −5.414294683736180, −5.045100169290938, −4.446676168785348, −4.180801083710884, −3.423893037652603, −2.890998653831359, −2.446631654670778, −1.791154883776012, −0.9147359481110744, −0.4536261517009635,
0.4536261517009635, 0.9147359481110744, 1.791154883776012, 2.446631654670778, 2.890998653831359, 3.423893037652603, 4.180801083710884, 4.446676168785348, 5.045100169290938, 5.414294683736180, 6.080423456098427, 6.725592016323210, 6.955469636222379, 7.559508322670770, 8.163857532378593, 8.529051396927011, 8.733361202292963, 9.185307555141827, 9.994869466876843, 10.57248010552454, 10.75604438376466, 11.45728680456608, 11.57360833379216, 11.99107481001686, 12.53587588026727
