L(s) = 1 | + i·2-s + 2.30i·3-s − 4-s − 2.30·6-s − 3.30i·7-s − i·8-s − 2.30·9-s − 2.60·11-s − 2.30i·12-s − 2.30i·13-s + 3.30·14-s + 16-s − 1.30i·17-s − 2.30i·18-s + 0.605·19-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + 1.32i·3-s − 0.5·4-s − 0.940·6-s − 1.24i·7-s − 0.353i·8-s − 0.767·9-s − 0.785·11-s − 0.664i·12-s − 0.638i·13-s + 0.882·14-s + 0.250·16-s − 0.315i·17-s − 0.542i·18-s + 0.138·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.288977683\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.288977683\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 29 | \( 1 - T \) |
good | 3 | \( 1 - 2.30iT - 3T^{2} \) |
| 7 | \( 1 + 3.30iT - 7T^{2} \) |
| 11 | \( 1 + 2.60T + 11T^{2} \) |
| 13 | \( 1 + 2.30iT - 13T^{2} \) |
| 17 | \( 1 + 1.30iT - 17T^{2} \) |
| 19 | \( 1 - 0.605T + 19T^{2} \) |
| 23 | \( 1 + 6.90iT - 23T^{2} \) |
| 31 | \( 1 - 6.69T + 31T^{2} \) |
| 37 | \( 1 - 0.605iT - 37T^{2} \) |
| 41 | \( 1 - 8.60T + 41T^{2} \) |
| 43 | \( 1 - 3.30iT - 43T^{2} \) |
| 47 | \( 1 + 5.21iT - 47T^{2} \) |
| 53 | \( 1 + 13.3iT - 53T^{2} \) |
| 59 | \( 1 - 7.69T + 59T^{2} \) |
| 61 | \( 1 - 0.302T + 61T^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 + 5.21T + 71T^{2} \) |
| 73 | \( 1 + 13.1iT - 73T^{2} \) |
| 79 | \( 1 + 8.90T + 79T^{2} \) |
| 83 | \( 1 + 13.8iT - 83T^{2} \) |
| 89 | \( 1 + 7.81T + 89T^{2} \) |
| 97 | \( 1 - 16.1iT - 97T^{2} \) |
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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
−9.746142910942981255537647923854, −8.737870705268335747851716468297, −7.992513167165279271635507804093, −7.21525028974239912422654905259, −6.28809117995147038902524273314, −5.19447288724455406323611802440, −4.60099350056560205776620827954, −3.87780207179798395143191010039, −2.84628327092854890665428303424, −0.58035298103125132904273288199,
1.22269177486588105291891970859, 2.23690256114523326514541631464, 2.88057096306761775705235458345, 4.30985046753736545249123150432, 5.51029168995967170844649015246, 6.06757015189466619920515615657, 7.16532514817065480619782039599, 7.902875590155849822906108101110, 8.645356646842319644454204135040, 9.398732924632308284440635060547