L(s) = 1 | + (−0.366 − 0.633i)2-s + (0.732 − 1.26i)4-s + (−0.5 + 0.866i)5-s + (2.36 + 4.09i)7-s − 2.53·8-s + 0.732·10-s + (2.86 + 4.96i)11-s + (−0.732 + 1.26i)13-s + (1.73 − 3i)14-s + (−0.535 − 0.928i)16-s − 2.73·17-s + 4.46·19-s + (0.732 + 1.26i)20-s + (2.09 − 3.63i)22-s + (1.73 − 3i)23-s + ⋯ |
L(s) = 1 | + (−0.258 − 0.448i)2-s + (0.366 − 0.633i)4-s + (−0.223 + 0.387i)5-s + (0.894 + 1.54i)7-s − 0.896·8-s + 0.231·10-s + (0.864 + 1.49i)11-s + (−0.203 + 0.351i)13-s + (0.462 − 0.801i)14-s + (−0.133 − 0.232i)16-s − 0.662·17-s + 1.02·19-s + (0.163 + 0.283i)20-s + (0.447 − 0.774i)22-s + (0.361 − 0.625i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.173i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.984 - 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.34287 + 0.117486i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.34287 + 0.117486i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (0.5 - 0.866i)T \) |
good | 2 | \( 1 + (0.366 + 0.633i)T + (-1 + 1.73i)T^{2} \) |
| 7 | \( 1 + (-2.36 - 4.09i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.86 - 4.96i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (0.732 - 1.26i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 2.73T + 17T^{2} \) |
| 19 | \( 1 - 4.46T + 19T^{2} \) |
| 23 | \( 1 + (-1.73 + 3i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (1.59 + 2.76i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-1.5 + 2.59i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 2.73T + 37T^{2} \) |
| 41 | \( 1 + (-3.59 + 6.23i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (0.0980 + 0.169i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-4.36 - 7.56i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 6.73T + 53T^{2} \) |
| 59 | \( 1 + (-4.13 + 7.16i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (2 + 3.46i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (1.73 - 3i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 3.73T + 71T^{2} \) |
| 73 | \( 1 + 7.66T + 73T^{2} \) |
| 79 | \( 1 + (7.73 + 13.3i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (1.09 + 1.90i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 5.19T + 89T^{2} \) |
| 97 | \( 1 + (-4.83 - 8.36i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−11.53061023993282105443458891742, −10.41387427268553587915732248925, −9.389930527715667877384248154295, −8.920066587151520944654138723853, −7.52081781168786211895604479539, −6.54677493187342236951340282122, −5.50937241325768617849831822555, −4.45707098216171995792816046018, −2.58646768796043826120598385252, −1.78341626162607574448489394371,
1.06781007769637193604601039689, 3.27690572201446400781601964000, 4.16539898078527563276244983248, 5.53411138505939342050733129579, 6.85076901694255310397055641949, 7.50456106190513090222657709631, 8.358456598159535712604206707716, 9.071732631947064442978278181325, 10.47427028693480789911715830912, 11.38176000394801369503691463226