| L(s) = 1 | + (−0.263 − 0.184i)2-s + (−0.648 − 1.78i)4-s + (−2.21 + 0.303i)5-s + (2.07 + 4.46i)7-s + (−0.324 + 1.20i)8-s + (0.639 + 0.328i)10-s + (−0.0393 − 0.0469i)11-s + (0.0455 + 0.0650i)13-s + (0.274 − 1.55i)14-s + (−2.59 + 2.17i)16-s + (1.04 + 3.89i)17-s + (1.80 − 1.04i)19-s + (1.97 + 3.75i)20-s + (0.00171 + 0.0196i)22-s + (5.88 + 2.74i)23-s + ⋯ |
| L(s) = 1 | + (−0.186 − 0.130i)2-s + (−0.324 − 0.891i)4-s + (−0.990 + 0.135i)5-s + (0.786 + 1.68i)7-s + (−0.114 + 0.427i)8-s + (0.202 + 0.103i)10-s + (−0.0118 − 0.0141i)11-s + (0.0126 + 0.0180i)13-s + (0.0734 − 0.416i)14-s + (−0.649 + 0.544i)16-s + (0.252 + 0.943i)17-s + (0.413 − 0.238i)19-s + (0.442 + 0.838i)20-s + (0.000366 + 0.00418i)22-s + (1.22 + 0.572i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.614 - 0.788i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.614 - 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.834231 + 0.407659i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.834231 + 0.407659i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (2.21 - 0.303i)T \) |
| good | 2 | \( 1 + (0.263 + 0.184i)T + (0.684 + 1.87i)T^{2} \) |
| 7 | \( 1 + (-2.07 - 4.46i)T + (-4.49 + 5.36i)T^{2} \) |
| 11 | \( 1 + (0.0393 + 0.0469i)T + (-1.91 + 10.8i)T^{2} \) |
| 13 | \( 1 + (-0.0455 - 0.0650i)T + (-4.44 + 12.2i)T^{2} \) |
| 17 | \( 1 + (-1.04 - 3.89i)T + (-14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (-1.80 + 1.04i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-5.88 - 2.74i)T + (14.7 + 17.6i)T^{2} \) |
| 29 | \( 1 + (-1.24 - 7.06i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (-0.209 + 0.0761i)T + (23.7 - 19.9i)T^{2} \) |
| 37 | \( 1 + (6.33 - 1.69i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + (-2.22 - 0.391i)T + (38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (0.325 - 3.71i)T + (-42.3 - 7.46i)T^{2} \) |
| 47 | \( 1 + (1.26 - 0.591i)T + (30.2 - 36.0i)T^{2} \) |
| 53 | \( 1 + (8.67 + 8.67i)T + 53iT^{2} \) |
| 59 | \( 1 + (-0.888 - 0.745i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (6.77 + 2.46i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (-6.22 + 4.36i)T + (22.9 - 62.9i)T^{2} \) |
| 71 | \( 1 + (6.33 + 3.65i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-6.42 - 1.72i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (6.06 - 1.06i)T + (74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (-2.31 + 3.30i)T + (-28.3 - 77.9i)T^{2} \) |
| 89 | \( 1 + (-5.25 - 9.09i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-9.77 - 0.855i)T + (95.5 + 16.8i)T^{2} \) |
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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
−11.28871378942679964710146943840, −10.70679433251408105633687945367, −9.376248787485684635600637010976, −8.710821634443302180410049379179, −7.975687760026224894209477714972, −6.59185799268252065621177832184, −5.41264812116740501944617064730, −4.80175499314946979214509850296, −3.14315689902077941023913527746, −1.59683162780092834925182411527,
0.71254551061077797961176970918, 3.20556459029017564143259861049, 4.19129401191914600036615883728, 4.86265345972306198120598740155, 6.94535582874890783817218040396, 7.50131068280325477222844241052, 8.116233878904587376785976416781, 9.111603096432846649948779121716, 10.31487196383733775825983485937, 11.22719331618625844125161842270
