| L(s) = 1 | − 7-s − 6·11-s − 6·13-s + 2·17-s + 7·19-s + 8·23-s − 6·29-s − 9·31-s − 3·37-s + 10·41-s + 43-s − 2·47-s − 6·49-s + 2·53-s − 12·59-s + 3·61-s + 4·67-s + 12·71-s + 11·73-s + 6·77-s − 11·79-s − 6·83-s + 8·89-s + 6·91-s + 7·97-s + 10·101-s + 5·103-s + ⋯ |
| L(s) = 1 | − 0.377·7-s − 1.80·11-s − 1.66·13-s + 0.485·17-s + 1.60·19-s + 1.66·23-s − 1.11·29-s − 1.61·31-s − 0.493·37-s + 1.56·41-s + 0.152·43-s − 0.291·47-s − 6/7·49-s + 0.274·53-s − 1.56·59-s + 0.384·61-s + 0.488·67-s + 1.42·71-s + 1.28·73-s + 0.683·77-s − 1.23·79-s − 0.658·83-s + 0.847·89-s + 0.628·91-s + 0.710·97-s + 0.995·101-s + 0.492·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.175512734\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.175512734\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 7 T + p T^{2} \) | 1.19.ah |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 9 T + p T^{2} \) | 1.31.j |
| 37 | \( 1 + 3 T + p T^{2} \) | 1.37.d |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 3 T + p T^{2} \) | 1.61.ad |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 11 T + p T^{2} \) | 1.73.al |
| 79 | \( 1 + 11 T + p T^{2} \) | 1.79.l |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 - 8 T + p T^{2} \) | 1.89.ai |
| 97 | \( 1 - 7 T + p T^{2} \) | 1.97.ah |
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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
−7.86629617934848154041526001183, −7.46163064092310664936259227453, −7.08091028962897187246030565199, −5.77606585155551996573611189930, −5.21759031220592452264692876120, −4.82497811403578669610578189296, −3.42780671686453598388488886254, −2.90328879407899015649372124329, −2.04021077346179182401979224597, −0.55451079281509536544611972434,
0.55451079281509536544611972434, 2.04021077346179182401979224597, 2.90328879407899015649372124329, 3.42780671686453598388488886254, 4.82497811403578669610578189296, 5.21759031220592452264692876120, 5.77606585155551996573611189930, 7.08091028962897187246030565199, 7.46163064092310664936259227453, 7.86629617934848154041526001183
