| L(s) = 1 | − 2·4-s − 5·7-s − 2·13-s + 4·16-s + 7·19-s − 5·25-s + 10·28-s + 11·31-s + 11·37-s − 8·43-s + 18·49-s + 4·52-s + 61-s − 8·64-s + 11·67-s − 17·73-s − 14·76-s + 13·79-s + 10·91-s + 5·97-s + 10·100-s − 7·103-s + 19·109-s − 20·112-s + ⋯ |
| L(s) = 1 | − 4-s − 1.88·7-s − 0.554·13-s + 16-s + 1.60·19-s − 25-s + 1.88·28-s + 1.97·31-s + 1.80·37-s − 1.21·43-s + 18/7·49-s + 0.554·52-s + 0.128·61-s − 64-s + 1.34·67-s − 1.98·73-s − 1.60·76-s + 1.46·79-s + 1.04·91-s + 0.507·97-s + 100-s − 0.689·103-s + 1.81·109-s − 1.88·112-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8296809864\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8296809864\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 11 | \( 1 \) | |
| good | 2 | \( 1 + p T^{2} \) | 1.2.a |
| 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + 5 T + p T^{2} \) | 1.7.f |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 7 T + p T^{2} \) | 1.19.ah |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 11 T + p T^{2} \) | 1.31.al |
| 37 | \( 1 - 11 T + p T^{2} \) | 1.37.al |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - T + p T^{2} \) | 1.61.ab |
| 67 | \( 1 - 11 T + p T^{2} \) | 1.67.al |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 17 T + p T^{2} \) | 1.73.r |
| 79 | \( 1 - 13 T + p T^{2} \) | 1.79.an |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 - 5 T + p T^{2} \) | 1.97.af |
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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
−9.738044712390117652477601253453, −9.386464349808574579623744962914, −8.277413898553324837040762491191, −7.40888065910781063604227830543, −6.40158080143163907333611081505, −5.65272361057678955740931295147, −4.59633679987356442004935711420, −3.57224945195733786318502793376, −2.81030152638481090628965604169, −0.68442490173741656332230083278,
0.68442490173741656332230083278, 2.81030152638481090628965604169, 3.57224945195733786318502793376, 4.59633679987356442004935711420, 5.65272361057678955740931295147, 6.40158080143163907333611081505, 7.40888065910781063604227830543, 8.277413898553324837040762491191, 9.386464349808574579623744962914, 9.738044712390117652477601253453
