L(s) = 1 | + i·2-s + i·3-s − 4-s + (2 − i)5-s − 6-s − i·8-s − 9-s + (1 + 2i)10-s + 2·11-s − i·12-s − 6i·13-s + (1 + 2i)15-s + 16-s + 2i·17-s − i·18-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + 0.577i·3-s − 0.5·4-s + (0.894 − 0.447i)5-s − 0.408·6-s − 0.353i·8-s − 0.333·9-s + (0.316 + 0.632i)10-s + 0.603·11-s − 0.288i·12-s − 1.66i·13-s + (0.258 + 0.516i)15-s + 0.250·16-s + 0.485i·17-s − 0.235i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.891417695\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.891417695\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-2 + i)T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 + 6iT - 13T^{2} \) |
| 17 | \( 1 - 2iT - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 4iT - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 + 8iT - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 - 10T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 - 8iT - 67T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 - 4iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 4iT - 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 + 8iT - 97T^{2} \) |
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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
−9.593148554494952185696762801018, −8.534156466492753092464422334564, −8.313272395432225578297641864524, −7.01389206298808619852366448788, −6.12271611463750799361807268494, −5.51024514190473386978132584240, −4.74496370413980754209600536122, −3.73917199616717111911288681286, −2.53322819024110257694410010583, −0.865571406357834911579609005348,
1.30261018380362140928162292080, 2.12487353228441756462978713389, 3.11398472898885359095516035133, 4.26985893026235069627115502598, 5.26316828814816728971688627562, 6.39644751423259458431223566533, 6.77572921875698048847123232792, 7.88094156907897375622247621512, 8.955552114257713804099780591386, 9.477712427087874597735074880976