| L(s) = 1 | + 2-s + 4-s + 4·5-s − 2·7-s + 8-s + 4·10-s − 6·13-s − 2·14-s + 16-s + 17-s + 4·19-s + 4·20-s − 6·23-s + 11·25-s − 6·26-s − 2·28-s + 4·29-s − 6·31-s + 32-s + 34-s − 8·35-s − 4·37-s + 4·38-s + 4·40-s + 10·41-s − 4·43-s − 6·46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 1.78·5-s − 0.755·7-s + 0.353·8-s + 1.26·10-s − 1.66·13-s − 0.534·14-s + 1/4·16-s + 0.242·17-s + 0.917·19-s + 0.894·20-s − 1.25·23-s + 11/5·25-s − 1.17·26-s − 0.377·28-s + 0.742·29-s − 1.07·31-s + 0.176·32-s + 0.171·34-s − 1.35·35-s − 0.657·37-s + 0.648·38-s + 0.632·40-s + 1.56·41-s − 0.609·43-s − 0.884·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 306 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 306 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.249192179\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.249192179\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 17 | \( 1 - T \) | |
| good | 5 | \( 1 - 4 T + p T^{2} \) | 1.5.ae |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 31 | \( 1 + 6 T + p T^{2} \) | 1.31.g |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + 4 T + p T^{2} \) | 1.61.e |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
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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
−12.08855787871916423938091855149, −10.59451577638683826251692098584, −9.809680223299837765262582433989, −9.330186221363971309576111465163, −7.58460004774890445683046626123, −6.52498883063792512928187502617, −5.71929507199710618712172159386, −4.84625070630626219217033988858, −3.08824258991321588799800323044, −2.01050304782327789083256558357,
2.01050304782327789083256558357, 3.08824258991321588799800323044, 4.84625070630626219217033988858, 5.71929507199710618712172159386, 6.52498883063792512928187502617, 7.58460004774890445683046626123, 9.330186221363971309576111465163, 9.809680223299837765262582433989, 10.59451577638683826251692098584, 12.08855787871916423938091855149
