| L(s) = 1 | − 3·5-s + 2·7-s + 6·11-s + 5·13-s + 3·17-s + 2·19-s − 6·23-s + 4·25-s − 3·29-s − 4·31-s − 6·35-s + 5·37-s + 6·41-s − 10·43-s − 3·49-s + 6·53-s − 18·55-s + 12·59-s + 5·61-s − 15·65-s + 2·67-s − 6·71-s − 73-s + 12·77-s − 10·79-s − 9·85-s + 3·89-s + ⋯ |
| L(s) = 1 | − 1.34·5-s + 0.755·7-s + 1.80·11-s + 1.38·13-s + 0.727·17-s + 0.458·19-s − 1.25·23-s + 4/5·25-s − 0.557·29-s − 0.718·31-s − 1.01·35-s + 0.821·37-s + 0.937·41-s − 1.52·43-s − 3/7·49-s + 0.824·53-s − 2.42·55-s + 1.56·59-s + 0.640·61-s − 1.86·65-s + 0.244·67-s − 0.712·71-s − 0.117·73-s + 1.36·77-s − 1.12·79-s − 0.976·85-s + 0.317·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.283583529\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.283583529\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 3 | \( 1 \) | |
| good | 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 13 | \( 1 - 5 T + p T^{2} \) | 1.13.af |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 3 T + p T^{2} \) | 1.29.d |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 5 T + p T^{2} \) | 1.37.af |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 5 T + p T^{2} \) | 1.61.af |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 + T + p T^{2} \) | 1.73.b |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 3 T + p T^{2} \) | 1.89.ad |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−11.59835841211344372341029742363, −11.07810635226277574253797453362, −9.677528525390612960317867289160, −8.579573961859898822258311157623, −7.949251102855060400602455037247, −6.89563277027074651963402661567, −5.72299699566267407044406059705, −4.13912411526585097190750656835, −3.66391361115582894073101159026, −1.33745094888120597007437567736,
1.33745094888120597007437567736, 3.66391361115582894073101159026, 4.13912411526585097190750656835, 5.72299699566267407044406059705, 6.89563277027074651963402661567, 7.949251102855060400602455037247, 8.579573961859898822258311157623, 9.677528525390612960317867289160, 11.07810635226277574253797453362, 11.59835841211344372341029742363
