| L(s) = 1 | − 2-s − 4-s + 3·8-s − 3·9-s − 3·11-s + 13-s − 16-s − 7·17-s + 3·18-s − 7·19-s + 3·22-s + 6·23-s − 26-s − 5·29-s − 5·32-s + 7·34-s + 3·36-s − 8·37-s + 7·38-s − 2·43-s + 3·44-s − 6·46-s − 7·47-s − 52-s + 3·53-s + 5·58-s − 7·59-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 1.06·8-s − 9-s − 0.904·11-s + 0.277·13-s − 1/4·16-s − 1.69·17-s + 0.707·18-s − 1.60·19-s + 0.639·22-s + 1.25·23-s − 0.196·26-s − 0.928·29-s − 0.883·32-s + 1.20·34-s + 1/2·36-s − 1.31·37-s + 1.13·38-s − 0.304·43-s + 0.452·44-s − 0.884·46-s − 1.02·47-s − 0.138·52-s + 0.412·53-s + 0.656·58-s − 0.911·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | 5 | \( 1 \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 2 | \( 1 + T + p T^{2} \) | 1.2.b |
| 3 | \( 1 + p T^{2} \) | 1.3.a |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 17 | \( 1 + 7 T + p T^{2} \) | 1.17.h |
| 19 | \( 1 + 7 T + p T^{2} \) | 1.19.h |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 5 T + p T^{2} \) | 1.29.f |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 + 7 T + p T^{2} \) | 1.47.h |
| 53 | \( 1 - 3 T + p T^{2} \) | 1.53.ad |
| 59 | \( 1 + 7 T + p T^{2} \) | 1.59.h |
| 61 | \( 1 + 7 T + p T^{2} \) | 1.61.h |
| 67 | \( 1 - 3 T + p T^{2} \) | 1.67.ad |
| 71 | \( 1 + 5 T + p T^{2} \) | 1.71.f |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 + 6 T + p T^{2} \) | 1.79.g |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−16.74312990585392, −16.08043443082615, −15.38179615009993, −14.94308052391517, −14.36853510939713, −13.57216693157896, −13.18361830429197, −12.91745948874674, −11.99471764637602, −11.17902668948490, −10.75864881357139, −10.52959164662607, −9.560367207549163, −8.945824040680706, −8.627928950388395, −8.202904059111420, −7.384946235278394, −6.759577171179702, −6.045279016740420, −5.246634122129710, −4.718336116905481, −4.045372263128988, −3.117600263312535, −2.321011863739740, −1.541706333034485, 0, 0,
1.541706333034485, 2.321011863739740, 3.117600263312535, 4.045372263128988, 4.718336116905481, 5.246634122129710, 6.045279016740420, 6.759577171179702, 7.384946235278394, 8.202904059111420, 8.627928950388395, 8.945824040680706, 9.560367207549163, 10.52959164662607, 10.75864881357139, 11.17902668948490, 11.99471764637602, 12.91745948874674, 13.18361830429197, 13.57216693157896, 14.36853510939713, 14.94308052391517, 15.38179615009993, 16.08043443082615, 16.74312990585392
