| L(s) = 1 | − 2·3-s + 9-s + 6·17-s + 6·19-s + 23-s − 5·25-s + 4·27-s + 6·29-s − 8·31-s − 2·37-s + 2·41-s + 8·43-s + 8·47-s − 12·51-s + 2·53-s − 12·57-s − 6·59-s + 12·67-s − 2·69-s − 8·71-s + 6·73-s + 10·75-s − 16·79-s − 11·81-s + 6·83-s − 12·87-s − 6·89-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1/3·9-s + 1.45·17-s + 1.37·19-s + 0.208·23-s − 25-s + 0.769·27-s + 1.11·29-s − 1.43·31-s − 0.328·37-s + 0.312·41-s + 1.21·43-s + 1.16·47-s − 1.68·51-s + 0.274·53-s − 1.58·57-s − 0.781·59-s + 1.46·67-s − 0.240·69-s − 0.949·71-s + 0.702·73-s + 1.15·75-s − 1.80·79-s − 1.22·81-s + 0.658·83-s − 1.28·87-s − 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72128 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) | |
| 7 | \( 1 \) | |
| 23 | \( 1 - T \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 5 | \( 1 + p T^{2} \) | 1.5.a |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + 16 T + p T^{2} \) | 1.79.q |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 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
−14.30018170940565, −13.94194877268016, −13.35883173306388, −12.57823003768520, −12.30815114757390, −11.84995648057372, −11.45147336763334, −10.85694504306973, −10.43789631794035, −9.904343714024007, −9.389110896002039, −8.869683342982183, −8.087057738595168, −7.561544730650321, −7.204895970426729, −6.467417920515486, −5.892451801430884, −5.377118233375271, −5.266622576925844, −4.336338899995144, −3.742421511352185, −3.080518481117086, −2.429033689004062, −1.342938119029799, −0.9264517522374728, 0,
0.9264517522374728, 1.342938119029799, 2.429033689004062, 3.080518481117086, 3.742421511352185, 4.336338899995144, 5.266622576925844, 5.377118233375271, 5.892451801430884, 6.467417920515486, 7.204895970426729, 7.561544730650321, 8.087057738595168, 8.869683342982183, 9.389110896002039, 9.904343714024007, 10.43789631794035, 10.85694504306973, 11.45147336763334, 11.84995648057372, 12.30815114757390, 12.57823003768520, 13.35883173306388, 13.94194877268016, 14.30018170940565
