| L(s) = 1 | − 5-s + 4·7-s − 4·11-s − 3·17-s − 4·23-s − 4·25-s + 29-s + 4·31-s − 4·35-s − 3·37-s − 9·41-s + 8·43-s + 8·47-s + 9·49-s + 9·53-s + 4·55-s + 4·59-s + 7·61-s + 4·67-s + 8·71-s − 11·73-s − 16·77-s + 4·79-s + 3·85-s − 6·89-s − 2·97-s + 101-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.51·7-s − 1.20·11-s − 0.727·17-s − 0.834·23-s − 4/5·25-s + 0.185·29-s + 0.718·31-s − 0.676·35-s − 0.493·37-s − 1.40·41-s + 1.21·43-s + 1.16·47-s + 9/7·49-s + 1.23·53-s + 0.539·55-s + 0.520·59-s + 0.896·61-s + 0.488·67-s + 0.949·71-s − 1.28·73-s − 1.82·77-s + 0.450·79-s + 0.325·85-s − 0.635·89-s − 0.203·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - T + p T^{2} \) | 1.29.ab |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 3 T + p T^{2} \) | 1.37.d |
| 41 | \( 1 + 9 T + p T^{2} \) | 1.41.j |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 7 T + p T^{2} \) | 1.61.ah |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 11 T + p T^{2} \) | 1.73.l |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 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
−15.62724678002507, −15.27082469084662, −14.60791260651475, −14.04563797199693, −13.58053951245463, −13.11073106583359, −12.18802984043777, −11.95510898273513, −11.29599479211183, −10.84917402286077, −10.31078449490650, −9.778643739144224, −8.822727289607318, −8.325829360824869, −8.030513938778858, −7.379209810624817, −6.874106054342274, −5.842841208997796, −5.424267811787627, −4.716501614209195, −4.250550370483717, −3.542781508419966, −2.406678227874951, −2.125949527287178, −1.047239256428780, 0,
1.047239256428780, 2.125949527287178, 2.406678227874951, 3.542781508419966, 4.250550370483717, 4.716501614209195, 5.424267811787627, 5.842841208997796, 6.874106054342274, 7.379209810624817, 8.030513938778858, 8.325829360824869, 8.822727289607318, 9.778643739144224, 10.31078449490650, 10.84917402286077, 11.29599479211183, 11.95510898273513, 12.18802984043777, 13.11073106583359, 13.58053951245463, 14.04563797199693, 14.60791260651475, 15.27082469084662, 15.62724678002507
