| L(s) = 1 | − 2·7-s − 11-s + 2·17-s − 3·19-s + 4·23-s − 3·29-s − 31-s + 10·37-s − 41-s − 10·43-s − 10·47-s − 3·49-s + 6·53-s + 13·59-s + 10·61-s − 8·67-s + 9·71-s + 14·73-s + 2·77-s − 10·83-s − 13·89-s + 10·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 0.755·7-s − 0.301·11-s + 0.485·17-s − 0.688·19-s + 0.834·23-s − 0.557·29-s − 0.179·31-s + 1.64·37-s − 0.156·41-s − 1.52·43-s − 1.45·47-s − 3/7·49-s + 0.824·53-s + 1.69·59-s + 1.28·61-s − 0.977·67-s + 1.06·71-s + 1.63·73-s + 0.227·77-s − 1.09·83-s − 1.37·89-s + 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 129600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 129600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + T + p T^{2} \) | 1.11.b |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 3 T + p T^{2} \) | 1.19.d |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 3 T + p T^{2} \) | 1.29.d |
| 31 | \( 1 + T + p T^{2} \) | 1.31.b |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 + T + p T^{2} \) | 1.41.b |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 + 10 T + p T^{2} \) | 1.47.k |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 13 T + p T^{2} \) | 1.59.an |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 9 T + p T^{2} \) | 1.71.aj |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 10 T + p T^{2} \) | 1.83.k |
| 89 | \( 1 + 13 T + p T^{2} \) | 1.89.n |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.53250573449615, −13.18208382273263, −12.79952488174163, −12.51844191133154, −11.66477205527368, −11.34914187843178, −10.94965432162196, −10.13027130220199, −9.899643143991858, −9.532478463867708, −8.757200977999740, −8.410079680425194, −7.886827951897316, −7.215668368214269, −6.817800791788779, −6.273299140083970, −5.817371103974958, −5.058425565387347, −4.827478629950922, −3.784063045346867, −3.647549666170172, −2.813211525693050, −2.371774085580957, −1.563049148453538, −0.7751223755062531, 0,
0.7751223755062531, 1.563049148453538, 2.371774085580957, 2.813211525693050, 3.647549666170172, 3.784063045346867, 4.827478629950922, 5.058425565387347, 5.817371103974958, 6.273299140083970, 6.817800791788779, 7.215668368214269, 7.886827951897316, 8.410079680425194, 8.757200977999740, 9.532478463867708, 9.899643143991858, 10.13027130220199, 10.94965432162196, 11.34914187843178, 11.66477205527368, 12.51844191133154, 12.79952488174163, 13.18208382273263, 13.53250573449615
