| L(s) = 1 | − 2·5-s + 13-s + 2·17-s − 4·19-s − 25-s + 6·29-s + 2·37-s + 6·41-s + 12·43-s − 4·47-s + 6·53-s + 8·59-s − 2·61-s − 2·65-s − 4·67-s + 12·71-s + 14·73-s − 8·83-s − 4·85-s − 18·89-s + 8·95-s + 6·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.277·13-s + 0.485·17-s − 0.917·19-s − 1/5·25-s + 1.11·29-s + 0.328·37-s + 0.937·41-s + 1.82·43-s − 0.583·47-s + 0.824·53-s + 1.04·59-s − 0.256·61-s − 0.248·65-s − 0.488·67-s + 1.42·71-s + 1.63·73-s − 0.878·83-s − 0.433·85-s − 1.90·89-s + 0.820·95-s + 0.609·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 366912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 366912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.194424212\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.194424212\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 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 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 8 T + p T^{2} \) | 1.83.i |
| 89 | \( 1 + 18 T + p T^{2} \) | 1.89.s |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
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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
−12.56929168089445, −12.09509586062635, −11.59668014376348, −11.08981971084745, −10.87331118852964, −10.28188325030431, −9.801720579657514, −9.333531946575226, −8.802934224848514, −8.247981653018677, −8.035260776706693, −7.560804168739424, −6.882602410014452, −6.674080546533886, −5.872883366248348, −5.655296458608730, −4.910363750623265, −4.339959983262202, −4.007088142864142, −3.596369843727006, −2.783750576995708, −2.487937081359167, −1.692767937347931, −0.9261610236289186, −0.4675277735658308,
0.4675277735658308, 0.9261610236289186, 1.692767937347931, 2.487937081359167, 2.783750576995708, 3.596369843727006, 4.007088142864142, 4.339959983262202, 4.910363750623265, 5.655296458608730, 5.872883366248348, 6.674080546533886, 6.882602410014452, 7.560804168739424, 8.035260776706693, 8.247981653018677, 8.802934224848514, 9.333531946575226, 9.801720579657514, 10.28188325030431, 10.87331118852964, 11.08981971084745, 11.59668014376348, 12.09509586062635, 12.56929168089445
