| L(s) = 1 | + 2-s + 4-s + 2·5-s + 2·7-s + 8-s + 2·10-s + 2·13-s + 2·14-s + 16-s − 6·17-s + 4·19-s + 2·20-s − 6·23-s − 25-s + 2·26-s + 2·28-s + 2·29-s + 4·31-s + 32-s − 6·34-s + 4·35-s + 4·37-s + 4·38-s + 2·40-s + 2·41-s + 43-s − 6·46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.894·5-s + 0.755·7-s + 0.353·8-s + 0.632·10-s + 0.554·13-s + 0.534·14-s + 1/4·16-s − 1.45·17-s + 0.917·19-s + 0.447·20-s − 1.25·23-s − 1/5·25-s + 0.392·26-s + 0.377·28-s + 0.371·29-s + 0.718·31-s + 0.176·32-s − 1.02·34-s + 0.676·35-s + 0.657·37-s + 0.648·38-s + 0.316·40-s + 0.312·41-s + 0.152·43-s − 0.884·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 774 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 774 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.916488157\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.916488157\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 43 | \( 1 - T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 - 4 T + p T^{2} \) | 1.53.ae |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 + 12 T + p T^{2} \) | 1.61.m |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−10.38608088519526993171242305234, −9.578949280187494715478128887255, −8.556782627098180497435410748704, −7.68561391479336656512275985130, −6.52545003207311835858544421722, −5.88033841038659656444980730991, −4.90926507246187420270133288229, −4.03174056766814563085794086286, −2.60692344300556342507381167126, −1.59259064274091439944282632672,
1.59259064274091439944282632672, 2.60692344300556342507381167126, 4.03174056766814563085794086286, 4.90926507246187420270133288229, 5.88033841038659656444980730991, 6.52545003207311835858544421722, 7.68561391479336656512275985130, 8.556782627098180497435410748704, 9.578949280187494715478128887255, 10.38608088519526993171242305234
