| L(s) = 1 | − 2-s − 4-s + 2·7-s + 3·8-s − 2·11-s + 2·13-s − 2·14-s − 16-s + 2·17-s + 2·22-s − 2·26-s − 2·28-s − 6·29-s + 4·31-s − 5·32-s − 2·34-s − 2·37-s + 2·41-s − 10·43-s + 2·44-s − 3·49-s − 2·52-s + 10·53-s + 6·56-s + 6·58-s − 10·61-s − 4·62-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 0.755·7-s + 1.06·8-s − 0.603·11-s + 0.554·13-s − 0.534·14-s − 1/4·16-s + 0.485·17-s + 0.426·22-s − 0.392·26-s − 0.377·28-s − 1.11·29-s + 0.718·31-s − 0.883·32-s − 0.342·34-s − 0.328·37-s + 0.312·41-s − 1.52·43-s + 0.301·44-s − 3/7·49-s − 0.277·52-s + 1.37·53-s + 0.801·56-s + 0.787·58-s − 1.28·61-s − 0.508·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) | |
| 5 | \( 1 \) | |
| 19 | \( 1 \) | |
| good | 2 | \( 1 + T + p T^{2} \) | 1.2.b |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 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
−14.12713098049572, −13.75872918097783, −13.19573733155713, −12.94328687645469, −12.15204855355377, −11.65057624714383, −11.15946159955097, −10.56760920477084, −10.27002342545659, −9.701372613108112, −9.133584817638059, −8.656506365098428, −8.174712944687932, −7.779905386809312, −7.334783778837084, −6.628117198844750, −5.901429233249884, −5.296311219510765, −4.913296215664687, −4.278422902692066, −3.671504640540541, −3.031921333275917, −2.081615199082048, −1.549704607129944, −0.8508917345179708, 0,
0.8508917345179708, 1.549704607129944, 2.081615199082048, 3.031921333275917, 3.671504640540541, 4.278422902692066, 4.913296215664687, 5.296311219510765, 5.901429233249884, 6.628117198844750, 7.334783778837084, 7.779905386809312, 8.174712944687932, 8.656506365098428, 9.133584817638059, 9.701372613108112, 10.27002342545659, 10.56760920477084, 11.15946159955097, 11.65057624714383, 12.15204855355377, 12.94328687645469, 13.19573733155713, 13.75872918097783, 14.12713098049572
