| L(s) = 1 | + 2-s + i·7-s − 8-s − 9-s + 2i·11-s + i·14-s − 16-s − 18-s + 2i·22-s − i·23-s − 25-s + i·29-s + i·37-s + 43-s − i·46-s + ⋯ |
| L(s) = 1 | + 2-s + i·7-s − 8-s − 9-s + 2i·11-s + i·14-s − 16-s − 18-s + 2i·22-s − i·23-s − 25-s + i·29-s + i·37-s + 43-s − i·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.242 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.242 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.204979993\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.204979993\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - iT \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - T + T^{2} \) |
| 3 | \( 1 + T^{2} \) |
| 5 | \( 1 + T^{2} \) |
| 11 | \( 1 - 2iT - T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + iT - T^{2} \) |
| 29 | \( 1 - iT - T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 - iT - T^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 - T + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - T + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 - 2T + T^{2} \) |
| 71 | \( 1 - iT - T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 + iT - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + T^{2} \) |
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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
−9.469800725045294088729612138539, −8.832723859148665475823117250291, −8.092071221071548118990102198610, −6.97238761624274728072185599682, −6.19078800976963923847146187520, −5.38091449339192771825603484625, −4.82121295513783316141270730102, −3.95754402711918567805560388612, −2.81856631067020030256052247131, −2.12868072810180895896204509593,
0.59996307249461474359803808839, 2.58478021045391380285616403114, 3.60867770842301081708954649092, 3.90178870613364338403077124290, 5.20845267894984839089636153944, 5.81608582669140801274101981517, 6.34595855389530726790197778443, 7.59423752581011762415355742773, 8.316652322437544151995568309201, 9.034912984808118246749820743418
