| L(s) = 1 | + 2-s + 4-s + (2.23 − 1.41i)7-s + 8-s + 1.41i·11-s + 5.39·13-s + (2.23 − 1.41i)14-s + 16-s + 2.23i·17-s + 1.30i·19-s + 1.41i·22-s + 23-s + 5.39·26-s + (2.23 − 1.41i)28-s + 9.24i·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + (0.845 − 0.534i)7-s + 0.353·8-s + 0.426i·11-s + 1.49·13-s + (0.597 − 0.377i)14-s + 0.250·16-s + 0.542i·17-s + 0.300i·19-s + 0.301i·22-s + 0.208·23-s + 1.05·26-s + (0.422 − 0.267i)28-s + 1.71i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0722i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0722i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.614578695\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.614578695\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-2.23 + 1.41i)T \) |
| good | 11 | \( 1 - 1.41iT - 11T^{2} \) |
| 13 | \( 1 - 5.39T + 13T^{2} \) |
| 17 | \( 1 - 2.23iT - 17T^{2} \) |
| 19 | \( 1 - 1.30iT - 19T^{2} \) |
| 23 | \( 1 - T + 23T^{2} \) |
| 29 | \( 1 - 9.24iT - 29T^{2} \) |
| 31 | \( 1 + 8.56iT - 31T^{2} \) |
| 37 | \( 1 - 2.82iT - 37T^{2} \) |
| 41 | \( 1 + 4.08T + 41T^{2} \) |
| 43 | \( 1 - 6.41iT - 43T^{2} \) |
| 47 | \( 1 + 7.63iT - 47T^{2} \) |
| 53 | \( 1 + 6.07T + 53T^{2} \) |
| 59 | \( 1 - 11.7T + 59T^{2} \) |
| 61 | \( 1 + 5.39iT - 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 4.34iT - 71T^{2} \) |
| 73 | \( 1 - 5.01T + 73T^{2} \) |
| 79 | \( 1 + 1.07T + 79T^{2} \) |
| 83 | \( 1 - 8.01iT - 83T^{2} \) |
| 89 | \( 1 - 15.2T + 89T^{2} \) |
| 97 | \( 1 + 18.4T + 97T^{2} \) |
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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
−8.419169690807411323745121858957, −8.011108427828764391562991249583, −7.04956028372084997325546588829, −6.41013100060320619263303871012, −5.54531945343789057849775741719, −4.80587964078635846851048088993, −3.96736676612888788265816003989, −3.37411254737377531120504563631, −2.00435793117497569964636886977, −1.19539301993949270472342004900,
1.07736682844406260445390677440, 2.16826248395092346880336251176, 3.14587158515161499169781840281, 4.00462302394114745083762719467, 4.86227924010914372572574346220, 5.59665449539923228691339021638, 6.22524189435961547558596296052, 7.05875476616026521591052489914, 7.996474276319849618718348697856, 8.573141849872687933813664921048
