| L(s) = 1 | − 2·7-s − 11-s + 6·13-s − 4·17-s + 2·19-s − 8·23-s − 6·37-s + 10·43-s − 3·49-s − 14·53-s − 12·59-s + 14·61-s + 4·67-s − 6·73-s + 2·77-s + 2·79-s − 16·83-s + 14·89-s − 12·91-s + 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + 8·119-s + ⋯ |
| L(s) = 1 | − 0.755·7-s − 0.301·11-s + 1.66·13-s − 0.970·17-s + 0.458·19-s − 1.66·23-s − 0.986·37-s + 1.52·43-s − 3/7·49-s − 1.92·53-s − 1.56·59-s + 1.79·61-s + 0.488·67-s − 0.702·73-s + 0.227·77-s + 0.225·79-s − 1.75·83-s + 1.48·89-s − 1.25·91-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 + 0.733·119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 158400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 158400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.163017097\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.163017097\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 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 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 14 T + p T^{2} \) | 1.53.o |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 - 2 T + p T^{2} \) | 1.79.ac |
| 83 | \( 1 + 16 T + p T^{2} \) | 1.83.q |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−13.22351219658447, −12.94521470763033, −12.36858065457872, −11.90892072452170, −11.26821247770575, −10.95287439681612, −10.49414449726519, −9.866685024673940, −9.540797504009993, −8.869900571679986, −8.565534336845410, −7.981773337768273, −7.503879259461153, −6.848545842315174, −6.265055038826512, −6.072919764411914, −5.517147923737804, −4.726619175809932, −4.231939436487807, −3.573859987220961, −3.311352886630987, −2.486156890270892, −1.881300200086386, −1.219180727537219, −0.3225212395025435,
0.3225212395025435, 1.219180727537219, 1.881300200086386, 2.486156890270892, 3.311352886630987, 3.573859987220961, 4.231939436487807, 4.726619175809932, 5.517147923737804, 6.072919764411914, 6.265055038826512, 6.848545842315174, 7.503879259461153, 7.981773337768273, 8.565534336845410, 8.869900571679986, 9.540797504009993, 9.866685024673940, 10.49414449726519, 10.95287439681612, 11.26821247770575, 11.90892072452170, 12.36858065457872, 12.94521470763033, 13.22351219658447
