L(s) = 1 | + (1.22 − 1.22i)3-s + (3.16 − 3.16i)7-s + 11-s + (−3.16 − 3.16i)13-s + (3.67 + 3.67i)17-s − 3i·19-s − 7.74i·21-s + (3.67 + 3.67i)27-s − 7.74·29-s + (1.22 − 1.22i)33-s + (3.16 − 3.16i)37-s − 7.74·39-s − 41-s + (−2.44 + 2.44i)43-s + (3.16 − 3.16i)47-s + ⋯ |
L(s) = 1 | + (0.707 − 0.707i)3-s + (1.19 − 1.19i)7-s + 0.301·11-s + (−0.877 − 0.877i)13-s + (0.891 + 0.891i)17-s − 0.688i·19-s − 1.69i·21-s + (0.707 + 0.707i)27-s − 1.43·29-s + (0.213 − 0.213i)33-s + (0.519 − 0.519i)37-s − 1.24·39-s − 0.156·41-s + (−0.373 + 0.373i)43-s + (0.461 − 0.461i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.320 + 0.947i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.320 + 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.76185 - 1.26345i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.76185 - 1.26345i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + (-1.22 + 1.22i)T - 3iT^{2} \) |
| 7 | \( 1 + (-3.16 + 3.16i)T - 7iT^{2} \) |
| 11 | \( 1 - T + 11T^{2} \) |
| 13 | \( 1 + (3.16 + 3.16i)T + 13iT^{2} \) |
| 17 | \( 1 + (-3.67 - 3.67i)T + 17iT^{2} \) |
| 19 | \( 1 + 3iT - 19T^{2} \) |
| 23 | \( 1 + 23iT^{2} \) |
| 29 | \( 1 + 7.74T + 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 + (-3.16 + 3.16i)T - 37iT^{2} \) |
| 41 | \( 1 + T + 41T^{2} \) |
| 43 | \( 1 + (2.44 - 2.44i)T - 43iT^{2} \) |
| 47 | \( 1 + (-3.16 + 3.16i)T - 47iT^{2} \) |
| 53 | \( 1 + (-6.32 - 6.32i)T + 53iT^{2} \) |
| 59 | \( 1 + 4iT - 59T^{2} \) |
| 61 | \( 1 - 7.74iT - 61T^{2} \) |
| 67 | \( 1 + (3.67 + 3.67i)T + 67iT^{2} \) |
| 71 | \( 1 + 7.74iT - 71T^{2} \) |
| 73 | \( 1 + (1.22 - 1.22i)T - 73iT^{2} \) |
| 79 | \( 1 + 7.74T + 79T^{2} \) |
| 83 | \( 1 + (-1.22 + 1.22i)T - 83iT^{2} \) |
| 89 | \( 1 - 13iT - 89T^{2} \) |
| 97 | \( 1 + (-4.89 - 4.89i)T + 97iT^{2} \) |
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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
−10.26385777844525643512890054032, −9.095339989718586397697401844715, −8.063141712548468710887900859502, −7.64985114075378748084400201289, −7.05466133986841275368939790045, −5.60343020776571734390026519466, −4.63607536514315794952997491495, −3.54566180329874361724599407920, −2.22142348442907391212073185960, −1.12070698651804185382596363228,
1.81546517934524829516181440016, 2.88244913855540096920653180934, 4.07829322438949224997724746420, 4.98685519888636699894701021180, 5.81536943916169453150608173382, 7.17405126266513217019158034920, 8.076043676548145614692703903458, 8.883915527116000359241161965334, 9.450395983018096397811377532694, 10.15819065430603124338223791442