| L(s) = 1 | + 2-s + 4-s − 7-s + 8-s + 6·11-s − 13-s − 14-s + 16-s + 6·17-s + 5·19-s + 6·22-s − 6·23-s − 5·25-s − 26-s − 28-s + 6·29-s − 7·31-s + 32-s + 6·34-s − 7·37-s + 5·38-s + 5·43-s + 6·44-s − 6·46-s − 6·47-s − 6·49-s − 5·50-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.377·7-s + 0.353·8-s + 1.80·11-s − 0.277·13-s − 0.267·14-s + 1/4·16-s + 1.45·17-s + 1.14·19-s + 1.27·22-s − 1.25·23-s − 25-s − 0.196·26-s − 0.188·28-s + 1.11·29-s − 1.25·31-s + 0.176·32-s + 1.02·34-s − 1.15·37-s + 0.811·38-s + 0.762·43-s + 0.904·44-s − 0.884·46-s − 0.875·47-s − 6/7·49-s − 0.707·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 486 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 486 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.266323598\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.266323598\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 7 T + p T^{2} \) | 1.31.h |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 5 T + p T^{2} \) | 1.43.af |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - 11 T + p T^{2} \) | 1.79.al |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + 12 T + p T^{2} \) | 1.89.m |
| 97 | \( 1 + 7 T + p T^{2} \) | 1.97.h |
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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
−11.24491673338291489731146429192, −9.943813929580416197080231374652, −9.462532112195837169722009684835, −8.119718303065574933176111998816, −7.13851164803832070932827069270, −6.23082135950629212838366670654, −5.36559149920159416821901753580, −4.02055663905092829594123573049, −3.28066431744467752166245100129, −1.54349533627005461870574872570,
1.54349533627005461870574872570, 3.28066431744467752166245100129, 4.02055663905092829594123573049, 5.36559149920159416821901753580, 6.23082135950629212838366670654, 7.13851164803832070932827069270, 8.119718303065574933176111998816, 9.462532112195837169722009684835, 9.943813929580416197080231374652, 11.24491673338291489731146429192
