| L(s) = 1 | + (1.18 − 1.31i)2-s + (−2.83 + 1.26i)3-s + (−0.120 − 1.14i)4-s + (0.978 − 0.207i)5-s + (−1.70 + 5.23i)6-s + (−2.35 − 1.21i)7-s + (1.22 + 0.887i)8-s + (4.44 − 4.93i)9-s + (0.887 − 1.53i)10-s + (2.89 + 1.62i)11-s + (1.78 + 3.08i)12-s + (1.56 + 4.81i)13-s + (−4.39 + 1.66i)14-s + (−2.51 + 1.82i)15-s + (4.86 − 1.03i)16-s + (4.54 + 5.04i)17-s + ⋯ |
| L(s) = 1 | + (0.839 − 0.932i)2-s + (−1.63 + 0.728i)3-s + (−0.0600 − 0.571i)4-s + (0.437 − 0.0929i)5-s + (−0.695 + 2.13i)6-s + (−0.888 − 0.458i)7-s + (0.431 + 0.313i)8-s + (1.48 − 1.64i)9-s + (0.280 − 0.486i)10-s + (0.872 + 0.489i)11-s + (0.514 + 0.891i)12-s + (0.434 + 1.33i)13-s + (−1.17 + 0.443i)14-s + (−0.648 + 0.471i)15-s + (1.21 − 0.258i)16-s + (1.10 + 1.22i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0492i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0492i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.37885 - 0.0339603i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.37885 - 0.0339603i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-0.978 + 0.207i)T \) |
| 7 | \( 1 + (2.35 + 1.21i)T \) |
| 11 | \( 1 + (-2.89 - 1.62i)T \) |
| good | 2 | \( 1 + (-1.18 + 1.31i)T + (-0.209 - 1.98i)T^{2} \) |
| 3 | \( 1 + (2.83 - 1.26i)T + (2.00 - 2.22i)T^{2} \) |
| 13 | \( 1 + (-1.56 - 4.81i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (-4.54 - 5.04i)T + (-1.77 + 16.9i)T^{2} \) |
| 19 | \( 1 + (-0.470 + 4.47i)T + (-18.5 - 3.95i)T^{2} \) |
| 23 | \( 1 + (-1.37 - 2.37i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.53 + 1.11i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (2.48 + 0.527i)T + (28.3 + 12.6i)T^{2} \) |
| 37 | \( 1 + (1.98 + 0.885i)T + (24.7 + 27.4i)T^{2} \) |
| 41 | \( 1 + (7.04 + 5.11i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + 4.20T + 43T^{2} \) |
| 47 | \( 1 + (0.284 - 2.70i)T + (-45.9 - 9.77i)T^{2} \) |
| 53 | \( 1 + (-1.42 - 0.302i)T + (48.4 + 21.5i)T^{2} \) |
| 59 | \( 1 + (-0.957 - 9.11i)T + (-57.7 + 12.2i)T^{2} \) |
| 61 | \( 1 + (-6.57 + 1.39i)T + (55.7 - 24.8i)T^{2} \) |
| 67 | \( 1 + (-4.25 + 7.37i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (0.762 - 2.34i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (0.0333 + 0.317i)T + (-71.4 + 15.1i)T^{2} \) |
| 79 | \( 1 + (3.26 - 3.63i)T + (-8.25 - 78.5i)T^{2} \) |
| 83 | \( 1 + (-5.48 + 16.8i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (0.447 + 0.775i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (1.39 + 4.28i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−11.49825817696744519820211444458, −10.59745764069486189348119106467, −9.997968306444788252800602068005, −9.128603597936403829333670763141, −7.02874779784103338184254367802, −6.25222136566166291827219127127, −5.26464026396393434961845693865, −4.24406093164312695284960960374, −3.60865288188187301842493290056, −1.45626470953113970907504114475,
1.04712597689630173007706942085, 3.44238689321861325555354418275, 5.22855817514306626673588846232, 5.57651499640886454924084980702, 6.42261611781370409479212795735, 6.92985089497796217760494591379, 8.106279676760928745541852465948, 9.824070967115714934408090490954, 10.48675557652738552797951332197, 11.66923530067322492632116252798
