| L(s) = 1 | − 7-s + 3·11-s − 13-s − 6·17-s − 4·19-s − 3·23-s − 3·29-s − 5·31-s + 2·37-s + 3·41-s + 43-s − 9·47-s − 6·49-s − 6·53-s − 3·59-s + 13·61-s + 7·67-s + 12·71-s + 10·73-s − 3·77-s − 11·79-s + 9·83-s + 6·89-s + 91-s − 11·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 0.377·7-s + 0.904·11-s − 0.277·13-s − 1.45·17-s − 0.917·19-s − 0.625·23-s − 0.557·29-s − 0.898·31-s + 0.328·37-s + 0.468·41-s + 0.152·43-s − 1.31·47-s − 6/7·49-s − 0.824·53-s − 0.390·59-s + 1.66·61-s + 0.855·67-s + 1.42·71-s + 1.17·73-s − 0.341·77-s − 1.23·79-s + 0.987·83-s + 0.635·89-s + 0.104·91-s − 1.11·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 129600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 129600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7053062470\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7053062470\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| good | 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 + 3 T + p T^{2} \) | 1.29.d |
| 31 | \( 1 + 5 T + p T^{2} \) | 1.31.f |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 3 T + p T^{2} \) | 1.41.ad |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 + 9 T + p T^{2} \) | 1.47.j |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 - 13 T + p T^{2} \) | 1.61.an |
| 67 | \( 1 - 7 T + p T^{2} \) | 1.67.ah |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 + 11 T + p T^{2} \) | 1.79.l |
| 83 | \( 1 - 9 T + p T^{2} \) | 1.83.aj |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 11 T + p T^{2} \) | 1.97.l |
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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
−13.36406040450713, −13.02392154577639, −12.55733562951851, −12.15287958534314, −11.43567126368158, −11.01766855665153, −10.86648871789711, −9.899104461598598, −9.587906642122022, −9.234595651437390, −8.557305503059541, −8.207074981866293, −7.585618589624252, −6.841410023367574, −6.564797895861940, −6.212861028344573, −5.420870912584741, −4.908417429246209, −4.172522718060516, −3.914664621541222, −3.262677701795629, −2.377725071231478, −2.039088811724947, −1.285233188666022, −0.2493821336166846,
0.2493821336166846, 1.285233188666022, 2.039088811724947, 2.377725071231478, 3.262677701795629, 3.914664621541222, 4.172522718060516, 4.908417429246209, 5.420870912584741, 6.212861028344573, 6.564797895861940, 6.841410023367574, 7.585618589624252, 8.207074981866293, 8.557305503059541, 9.234595651437390, 9.587906642122022, 9.899104461598598, 10.86648871789711, 11.01766855665153, 11.43567126368158, 12.15287958534314, 12.55733562951851, 13.02392154577639, 13.36406040450713
