| L(s) = 1 | − 3-s + 5-s − 7-s + 9-s + 2·13-s − 15-s + 2·17-s + 21-s + 25-s − 27-s + 6·29-s + 4·31-s − 35-s − 2·37-s − 2·39-s + 10·41-s + 4·43-s + 45-s + 49-s − 2·51-s + 2·53-s + 4·59-s + 6·61-s − 63-s + 2·65-s − 12·67-s + 12·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 0.377·7-s + 1/3·9-s + 0.554·13-s − 0.258·15-s + 0.485·17-s + 0.218·21-s + 1/5·25-s − 0.192·27-s + 1.11·29-s + 0.718·31-s − 0.169·35-s − 0.328·37-s − 0.320·39-s + 1.56·41-s + 0.609·43-s + 0.149·45-s + 1/7·49-s − 0.280·51-s + 0.274·53-s + 0.520·59-s + 0.768·61-s − 0.125·63-s + 0.248·65-s − 1.46·67-s + 1.42·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.387878322\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.387878322\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 5 | \( 1 - T \) | |
| 7 | \( 1 + T \) | |
| good | 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 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
−10.25433762842440888827318619309, −9.489763666912341698118938418929, −8.582172802159158584080485405705, −7.56372664903441496690654081739, −6.53867770269247743908882645220, −5.92951446486079174583591122396, −4.96841409459491319087151202406, −3.86212103760575002233404144776, −2.59581724541211640514162451669, −1.03367082197206844928064846342,
1.03367082197206844928064846342, 2.59581724541211640514162451669, 3.86212103760575002233404144776, 4.96841409459491319087151202406, 5.92951446486079174583591122396, 6.53867770269247743908882645220, 7.56372664903441496690654081739, 8.582172802159158584080485405705, 9.489763666912341698118938418929, 10.25433762842440888827318619309
