| L(s) = 1 | + 7-s + 11-s + 4·13-s + 3·17-s + 5·19-s + 3·23-s + 6·29-s + 8·31-s + 7·37-s − 9·41-s − 8·43-s − 3·47-s − 6·49-s + 6·53-s − 3·59-s + 14·61-s − 2·67-s + 9·71-s − 2·73-s + 77-s − 79-s + 12·83-s − 18·89-s + 4·91-s − 11·97-s − 15·101-s + 4·103-s + ⋯ |
| L(s) = 1 | + 0.377·7-s + 0.301·11-s + 1.10·13-s + 0.727·17-s + 1.14·19-s + 0.625·23-s + 1.11·29-s + 1.43·31-s + 1.15·37-s − 1.40·41-s − 1.21·43-s − 0.437·47-s − 6/7·49-s + 0.824·53-s − 0.390·59-s + 1.79·61-s − 0.244·67-s + 1.06·71-s − 0.234·73-s + 0.113·77-s − 0.112·79-s + 1.31·83-s − 1.90·89-s + 0.419·91-s − 1.11·97-s − 1.49·101-s + 0.394·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.879559516\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.879559516\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| good | 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 7 T + p T^{2} \) | 1.37.ah |
| 41 | \( 1 + 9 T + p T^{2} \) | 1.41.j |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 - 9 T + p T^{2} \) | 1.71.aj |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + T + p T^{2} \) | 1.79.b |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 18 T + p T^{2} \) | 1.89.s |
| 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
−7.85774650825639680378676508382, −6.77269717487846203690995545403, −6.50271818222454114352519893816, −5.50081319599068715124578463993, −5.01905571009372582506627337197, −4.16113893051346796211758251657, −3.36512764911309893935966272925, −2.75090623647885184477030554324, −1.47432941596688366128662271423, −0.913667091627177150897834022864,
0.913667091627177150897834022864, 1.47432941596688366128662271423, 2.75090623647885184477030554324, 3.36512764911309893935966272925, 4.16113893051346796211758251657, 5.01905571009372582506627337197, 5.50081319599068715124578463993, 6.50271818222454114352519893816, 6.77269717487846203690995545403, 7.85774650825639680378676508382
