| L(s) = 1 | + 3·7-s + 2·11-s − 2·13-s − 5·17-s + 2·19-s − 8·23-s + 10·37-s − 3·41-s + 2·43-s − 9·47-s + 2·49-s − 12·53-s + 6·59-s + 2·61-s + 12·67-s − 12·71-s + 13·73-s + 6·77-s + 79-s − 4·83-s + 7·89-s − 6·91-s − 97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | + 1.13·7-s + 0.603·11-s − 0.554·13-s − 1.21·17-s + 0.458·19-s − 1.66·23-s + 1.64·37-s − 0.468·41-s + 0.304·43-s − 1.31·47-s + 2/7·49-s − 1.64·53-s + 0.781·59-s + 0.256·61-s + 1.46·67-s − 1.42·71-s + 1.52·73-s + 0.683·77-s + 0.112·79-s − 0.439·83-s + 0.741·89-s − 0.628·91-s − 0.101·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 5 T + p T^{2} \) | 1.17.f |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 + 3 T + p T^{2} \) | 1.41.d |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 + 9 T + p T^{2} \) | 1.47.j |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 - 13 T + p T^{2} \) | 1.73.an |
| 79 | \( 1 - T + p T^{2} \) | 1.79.ab |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 7 T + p T^{2} \) | 1.89.ah |
| 97 | \( 1 + T + p T^{2} \) | 1.97.b |
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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
−14.36502634795470, −14.11329234751998, −13.61217882949705, −12.83775903974353, −12.58232759810646, −11.66297987972294, −11.53597568761993, −11.16938117109846, −10.40004809758986, −9.852196146712372, −9.425562072556141, −8.827164718464334, −8.130960403324540, −7.935704487212299, −7.297650724057222, −6.525241719479057, −6.245310246846320, −5.423966617313792, −4.849826700749102, −4.389271738878075, −3.877565520281532, −3.057959473149872, −2.171671464717392, −1.867074289698749, −0.9862950757182533, 0,
0.9862950757182533, 1.867074289698749, 2.171671464717392, 3.057959473149872, 3.877565520281532, 4.389271738878075, 4.849826700749102, 5.423966617313792, 6.245310246846320, 6.525241719479057, 7.297650724057222, 7.935704487212299, 8.130960403324540, 8.827164718464334, 9.425562072556141, 9.852196146712372, 10.40004809758986, 11.16938117109846, 11.53597568761993, 11.66297987972294, 12.58232759810646, 12.83775903974353, 13.61217882949705, 14.11329234751998, 14.36502634795470
