| L(s) = 1 | − 7-s + 11-s + 17-s + 19-s − 5·23-s + 6·29-s − 6·31-s + 9·37-s + 5·41-s + 4·43-s − 7·47-s − 6·49-s + 4·53-s − 9·59-s − 12·61-s + 4·67-s − 15·71-s − 4·73-s − 77-s − 11·79-s + 6·83-s − 8·89-s + 97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | − 0.377·7-s + 0.301·11-s + 0.242·17-s + 0.229·19-s − 1.04·23-s + 1.11·29-s − 1.07·31-s + 1.47·37-s + 0.780·41-s + 0.609·43-s − 1.02·47-s − 6/7·49-s + 0.549·53-s − 1.17·59-s − 1.53·61-s + 0.488·67-s − 1.78·71-s − 0.468·73-s − 0.113·77-s − 1.23·79-s + 0.658·83-s − 0.847·89-s + 0.101·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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.b |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 - T + p T^{2} \) | 1.17.ab |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 + 5 T + p T^{2} \) | 1.23.f |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 6 T + p T^{2} \) | 1.31.g |
| 37 | \( 1 - 9 T + p T^{2} \) | 1.37.aj |
| 41 | \( 1 - 5 T + p T^{2} \) | 1.41.af |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 7 T + p T^{2} \) | 1.47.h |
| 53 | \( 1 - 4 T + p T^{2} \) | 1.53.ae |
| 59 | \( 1 + 9 T + p T^{2} \) | 1.59.j |
| 61 | \( 1 + 12 T + p T^{2} \) | 1.61.m |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 15 T + p T^{2} \) | 1.71.p |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 + 11 T + p T^{2} \) | 1.79.l |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + 8 T + p T^{2} \) | 1.89.i |
| 97 | \( 1 - T + p T^{2} \) | 1.97.ab |
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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
−16.03786226486191, −15.47777010691852, −14.73658007443912, −14.36835430277384, −13.81795604215069, −13.19408122435258, −12.62207038997642, −12.19728664949112, −11.50276647481075, −11.07603924125126, −10.27865548862114, −9.868535513858996, −9.258133294047520, −8.730175402597087, −7.936954762387724, −7.527109529167322, −6.802731167897023, −5.974094682953013, −5.874517918017530, −4.700280689859754, −4.322931394412948, −3.417669417641055, −2.867541820932547, −1.952786993241799, −1.097217893408254, 0,
1.097217893408254, 1.952786993241799, 2.867541820932547, 3.417669417641055, 4.322931394412948, 4.700280689859754, 5.874517918017530, 5.974094682953013, 6.802731167897023, 7.527109529167322, 7.936954762387724, 8.730175402597087, 9.258133294047520, 9.868535513858996, 10.27865548862114, 11.07603924125126, 11.50276647481075, 12.19728664949112, 12.62207038997642, 13.19408122435258, 13.81795604215069, 14.36835430277384, 14.73658007443912, 15.47777010691852, 16.03786226486191
