| L(s) = 1 | + (−3.67 + 2.12i)5-s + (3.5 + 6.06i)7-s + (11.0 + 6.36i)11-s − 13-s + (−14.6 − 8.48i)17-s + (11.5 + 19.9i)19-s + (−14.6 + 8.48i)23-s + (−3.5 + 6.06i)25-s − 33.9i·29-s + (23.5 − 40.7i)31-s + (−25.7 − 14.8i)35-s + (27.5 + 47.6i)37-s + 46.6i·41-s − 23·43-s + (−3.67 + 2.12i)47-s + ⋯ |
| L(s) = 1 | + (−0.734 + 0.424i)5-s + (0.5 + 0.866i)7-s + (1.00 + 0.578i)11-s − 0.0769·13-s + (−0.864 − 0.499i)17-s + (0.605 + 1.04i)19-s + (−0.638 + 0.368i)23-s + (−0.140 + 0.242i)25-s − 1.17i·29-s + (0.758 − 1.31i)31-s + (−0.734 − 0.424i)35-s + (0.743 + 1.28i)37-s + 1.13i·41-s − 0.534·43-s + (−0.0781 + 0.0451i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.778 - 0.627i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.778 - 0.627i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.141349126\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.141349126\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-3.5 - 6.06i)T \) |
| good | 5 | \( 1 + (3.67 - 2.12i)T + (12.5 - 21.6i)T^{2} \) |
| 11 | \( 1 + (-11.0 - 6.36i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + T + 169T^{2} \) |
| 17 | \( 1 + (14.6 + 8.48i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (-11.5 - 19.9i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (14.6 - 8.48i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + 33.9iT - 841T^{2} \) |
| 31 | \( 1 + (-23.5 + 40.7i)T + (-480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + (-27.5 - 47.6i)T + (-684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 - 46.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 23T + 1.84e3T^{2} \) |
| 47 | \( 1 + (3.67 - 2.12i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (44.0 + 25.4i)T + (1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (73.4 + 42.4i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (52 + 90.0i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (48.5 - 84.0i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 - 97.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (32.5 - 56.2i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-56.5 - 97.8i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + 29.6iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (117. - 67.8i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 - 104T + 9.40e3T^{2} \) |
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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
−9.789626527203213705509967029425, −9.491910949684273670104252582288, −8.177406358592894408457893426046, −7.82206763808125978357916089147, −6.67002207849982045744605004101, −5.93992675712439646958305689787, −4.71628310605330985062119529121, −3.94812296232787813094696104652, −2.74850016733247159906303725724, −1.56385783069821463230697734231,
0.37066812012022831688340842107, 1.53778401678891906041078390295, 3.21600918766239151894838956819, 4.22555762241875516712526811612, 4.76014893376309859007756981923, 6.12552813725567231487018178080, 7.01105009072913780921646206953, 7.76621777180683436286202351235, 8.698547163483840867832020388867, 9.176629307088286294501933770286
