L(s) = 1 | + (0.647 − 1.25i)2-s + (−1.78 + 1.78i)3-s + (−1.16 − 1.62i)4-s + (0.0829 + 2.23i)5-s + (1.09 + 3.40i)6-s + (−0.904 − 0.904i)7-s + (−2.79 + 0.408i)8-s − 3.39i·9-s + (2.86 + 1.34i)10-s − 4.92i·11-s + (4.98 + 0.831i)12-s + (−4.64 − 4.64i)13-s + (−1.72 + 0.551i)14-s + (−4.14 − 3.84i)15-s + (−1.29 + 3.78i)16-s + (−3.50 + 3.50i)17-s + ⋯ |
L(s) = 1 | + (0.457 − 0.889i)2-s + (−1.03 + 1.03i)3-s + (−0.581 − 0.813i)4-s + (0.0370 + 0.999i)5-s + (0.445 + 1.39i)6-s + (−0.341 − 0.341i)7-s + (−0.989 + 0.144i)8-s − 1.13i·9-s + (0.905 + 0.424i)10-s − 1.48i·11-s + (1.44 + 0.240i)12-s + (−1.28 − 1.28i)13-s + (−0.460 + 0.147i)14-s + (−1.07 − 0.993i)15-s + (−0.324 + 0.945i)16-s + (−0.850 + 0.850i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 + 0.103i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.994 + 0.103i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0120912 - 0.232823i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0120912 - 0.232823i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.647 + 1.25i)T \) |
| 5 | \( 1 + (-0.0829 - 2.23i)T \) |
| 19 | \( 1 + T \) |
good | 3 | \( 1 + (1.78 - 1.78i)T - 3iT^{2} \) |
| 7 | \( 1 + (0.904 + 0.904i)T + 7iT^{2} \) |
| 11 | \( 1 + 4.92iT - 11T^{2} \) |
| 13 | \( 1 + (4.64 + 4.64i)T + 13iT^{2} \) |
| 17 | \( 1 + (3.50 - 3.50i)T - 17iT^{2} \) |
| 23 | \( 1 + (-2.65 + 2.65i)T - 23iT^{2} \) |
| 29 | \( 1 + 4.23iT - 29T^{2} \) |
| 31 | \( 1 - 6.52iT - 31T^{2} \) |
| 37 | \( 1 + (4.68 - 4.68i)T - 37iT^{2} \) |
| 41 | \( 1 - 3.99T + 41T^{2} \) |
| 43 | \( 1 + (1.41 - 1.41i)T - 43iT^{2} \) |
| 47 | \( 1 + (0.801 + 0.801i)T + 47iT^{2} \) |
| 53 | \( 1 + (2.72 + 2.72i)T + 53iT^{2} \) |
| 59 | \( 1 + 13.9T + 59T^{2} \) |
| 61 | \( 1 - 6.61T + 61T^{2} \) |
| 67 | \( 1 + (1.41 + 1.41i)T + 67iT^{2} \) |
| 71 | \( 1 + 0.666iT - 71T^{2} \) |
| 73 | \( 1 + (-9.09 - 9.09i)T + 73iT^{2} \) |
| 79 | \( 1 + 7.09T + 79T^{2} \) |
| 83 | \( 1 + (6.54 - 6.54i)T - 83iT^{2} \) |
| 89 | \( 1 - 6.08iT - 89T^{2} \) |
| 97 | \( 1 + (-1.52 + 1.52i)T - 97iT^{2} \) |
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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
−10.78567584031038890976023261956, −10.43085946073292535752361092623, −9.709408966409797219559872694605, −8.361279652760057484084503241364, −6.62899551793009804755513310055, −5.81531607437407539466880185140, −4.90716624739772350683907882431, −3.72476134417638037783581788009, −2.81391656853781015160626588884, −0.14347159738999094511125820077,
2.08389051310091181830622862208, 4.48352538067230257780798785573, 5.03585259224905465616295249954, 6.13338695814091434752541318628, 7.11677782827995092049320310804, 7.47003019463606413880379289395, 9.087335460447535695853013560585, 9.528230738380507966166632667996, 11.44308037774028027588714446002, 12.19989959214451357590774093596