| L(s) = 1 | − i·2-s + i·3-s − 4-s + 6-s + 3i·7-s + i·8-s − 9-s − 5·11-s − i·12-s − 4i·13-s + 3·14-s + 16-s − i·17-s + i·18-s + 19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 0.577i·3-s − 0.5·4-s + 0.408·6-s + 1.13i·7-s + 0.353i·8-s − 0.333·9-s − 1.50·11-s − 0.288i·12-s − 1.10i·13-s + 0.801·14-s + 0.250·16-s − 0.242i·17-s + 0.235i·18-s + 0.229·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7318645621\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7318645621\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 17 | \( 1 + iT \) |
| good | 7 | \( 1 - 3iT - 7T^{2} \) |
| 11 | \( 1 + 5T + 11T^{2} \) |
| 13 | \( 1 + 4iT - 13T^{2} \) |
| 19 | \( 1 - T + 19T^{2} \) |
| 23 | \( 1 - 4iT - 23T^{2} \) |
| 29 | \( 1 - 4T + 29T^{2} \) |
| 31 | \( 1 + T + 31T^{2} \) |
| 37 | \( 1 + 9iT - 37T^{2} \) |
| 41 | \( 1 + 10T + 41T^{2} \) |
| 43 | \( 1 + 11iT - 43T^{2} \) |
| 47 | \( 1 - 9iT - 47T^{2} \) |
| 53 | \( 1 + 3iT - 53T^{2} \) |
| 59 | \( 1 - 8T + 59T^{2} \) |
| 61 | \( 1 + 14T + 61T^{2} \) |
| 67 | \( 1 + 7iT - 67T^{2} \) |
| 71 | \( 1 - 14T + 71T^{2} \) |
| 73 | \( 1 + 2iT - 73T^{2} \) |
| 79 | \( 1 - 5T + 79T^{2} \) |
| 83 | \( 1 + 8iT - 83T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 + 14iT - 97T^{2} \) |
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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
−8.758228940635798909519084726514, −8.122621505622242744883524592800, −7.36445461508955734346350956935, −5.90108256409752210919316025928, −5.36862531987375799769828732567, −4.84312607809245941682075205108, −3.51171618457979135614276262987, −2.87514947926548939877522874911, −2.06017842999979627194415357239, −0.26533985847052469959919698792,
1.10343457171104092569875563858, 2.44386134705013953532269830688, 3.58569093515897605389985210824, 4.62484371700703478911762445284, 5.18741852439797143034511018518, 6.43718923267881036432171038090, 6.75373804742397315424181690602, 7.65258197608749727134915894114, 8.120594977203527749528302513906, 8.872170636558553188906265883403
