| L(s) = 1 | + 7-s − 3·11-s − 2·13-s + 3·17-s − 4·19-s + 6·29-s + 8·31-s − 2·37-s + 43-s − 6·47-s − 6·49-s + 9·53-s − 12·59-s − 61-s + 7·67-s − 15·71-s + 4·73-s − 3·77-s − 10·79-s + 6·83-s − 2·91-s − 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 0.377·7-s − 0.904·11-s − 0.554·13-s + 0.727·17-s − 0.917·19-s + 1.11·29-s + 1.43·31-s − 0.328·37-s + 0.152·43-s − 0.875·47-s − 6/7·49-s + 1.23·53-s − 1.56·59-s − 0.128·61-s + 0.855·67-s − 1.78·71-s + 0.468·73-s − 0.341·77-s − 1.12·79-s + 0.658·83-s − 0.209·91-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| good | 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + T + p T^{2} \) | 1.61.b |
| 67 | \( 1 - 7 T + p T^{2} \) | 1.67.ah |
| 71 | \( 1 + 15 T + p T^{2} \) | 1.71.p |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 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
−15.53810736858667, −15.26192855061558, −14.50453487369823, −14.18325784745026, −13.51039937293250, −12.98515453077117, −12.40924859692881, −11.94582411924580, −11.37722359574517, −10.67629863539086, −10.14470392351649, −9.907605370668133, −8.938384925992238, −8.429502006358088, −7.897487704656704, −7.431957313890125, −6.615065543532566, −6.116179122439557, −5.329871925353316, −4.764163363240241, −4.315240376970662, −3.270115604880179, −2.721798410452255, −1.985534947362438, −1.046108779108452, 0,
1.046108779108452, 1.985534947362438, 2.721798410452255, 3.270115604880179, 4.315240376970662, 4.764163363240241, 5.329871925353316, 6.116179122439557, 6.615065543532566, 7.431957313890125, 7.897487704656704, 8.429502006358088, 8.938384925992238, 9.907605370668133, 10.14470392351649, 10.67629863539086, 11.37722359574517, 11.94582411924580, 12.40924859692881, 12.98515453077117, 13.51039937293250, 14.18325784745026, 14.50453487369823, 15.26192855061558, 15.53810736858667
