| L(s) = 1 | + 2-s − 2·3-s + 4-s + 5-s − 2·6-s + 2·7-s + 8-s + 9-s + 10-s − 2·12-s + 2·13-s + 2·14-s − 2·15-s + 16-s + 6·17-s + 18-s + 2·19-s + 20-s − 4·21-s − 2·24-s + 25-s + 2·26-s + 4·27-s + 2·28-s + 6·29-s − 2·30-s − 10·31-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.15·3-s + 1/2·4-s + 0.447·5-s − 0.816·6-s + 0.755·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.577·12-s + 0.554·13-s + 0.534·14-s − 0.516·15-s + 1/4·16-s + 1.45·17-s + 0.235·18-s + 0.458·19-s + 0.223·20-s − 0.872·21-s − 0.408·24-s + 1/5·25-s + 0.392·26-s + 0.769·27-s + 0.377·28-s + 1.11·29-s − 0.365·30-s − 1.79·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.680601593\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.680601593\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 5 | \( 1 - T \) | |
| 37 | \( 1 - T \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 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.ag |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−11.51689509480073388183484182312, −10.75944664966930507296718388412, −9.930390090643794017993197907992, −8.508271342304997015549036026098, −7.37364372891229977185623115606, −6.24527726803627967890931657857, −5.49324095653367495167136231203, −4.80762486474812590647702092154, −3.29880100176274410685427430205, −1.44505436450462837187908934432,
1.44505436450462837187908934432, 3.29880100176274410685427430205, 4.80762486474812590647702092154, 5.49324095653367495167136231203, 6.24527726803627967890931657857, 7.37364372891229977185623115606, 8.508271342304997015549036026098, 9.930390090643794017993197907992, 10.75944664966930507296718388412, 11.51689509480073388183484182312
