| L(s) = 1 | − 2-s + 4-s − 8-s + 3·13-s + 16-s − 7·17-s − 6·19-s − 5·23-s − 5·25-s − 3·26-s + 3·29-s − 7·31-s − 32-s + 7·34-s − 2·37-s + 6·38-s − 2·41-s + 13·43-s + 5·46-s + 6·47-s + 5·50-s + 3·52-s + 7·53-s − 3·58-s − 3·59-s + 10·61-s + 7·62-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s + 0.832·13-s + 1/4·16-s − 1.69·17-s − 1.37·19-s − 1.04·23-s − 25-s − 0.588·26-s + 0.557·29-s − 1.25·31-s − 0.176·32-s + 1.20·34-s − 0.328·37-s + 0.973·38-s − 0.312·41-s + 1.98·43-s + 0.737·46-s + 0.875·47-s + 0.707·50-s + 0.416·52-s + 0.961·53-s − 0.393·58-s − 0.390·59-s + 1.28·61-s + 0.889·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320166 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320166 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6053477973\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6053477973\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 3 | \( 1 \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 13 | \( 1 - 3 T + p T^{2} \) | 1.13.ad |
| 17 | \( 1 + 7 T + p T^{2} \) | 1.17.h |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 + 5 T + p T^{2} \) | 1.23.f |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 + 7 T + p T^{2} \) | 1.31.h |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 13 T + p T^{2} \) | 1.43.an |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 - 7 T + p T^{2} \) | 1.53.ah |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 13 T + p T^{2} \) | 1.67.n |
| 71 | \( 1 - 5 T + p T^{2} \) | 1.71.af |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + 14 T + p T^{2} \) | 1.79.o |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 15 T + p T^{2} \) | 1.89.ap |
| 97 | \( 1 - 12 T + p T^{2} \) | 1.97.am |
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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
−12.52424904741644, −12.17685626717159, −11.56727745770649, −11.11231762600652, −10.86096990063862, −10.29724607312005, −10.03818237187661, −9.217081090441393, −8.899088513361457, −8.653264377442880, −8.132466185955786, −7.494851722592146, −7.202141244601893, −6.504422947752728, −6.120361855787898, −5.859141755071044, −5.096461487878030, −4.349589024142432, −4.009641392722727, −3.610248667342569, −2.657383975942629, −2.141700476822602, −1.911992719230427, −1.026076766339035, −0.2467589206967609,
0.2467589206967609, 1.026076766339035, 1.911992719230427, 2.141700476822602, 2.657383975942629, 3.610248667342569, 4.009641392722727, 4.349589024142432, 5.096461487878030, 5.859141755071044, 6.120361855787898, 6.504422947752728, 7.202141244601893, 7.494851722592146, 8.132466185955786, 8.653264377442880, 8.899088513361457, 9.217081090441393, 10.03818237187661, 10.29724607312005, 10.86096990063862, 11.11231762600652, 11.56727745770649, 12.17685626717159, 12.52424904741644
