L(s) = 1 | − i·2-s + i·3-s − 4-s + 6-s − 2i·7-s + i·8-s − 9-s + 6·11-s − i·12-s + 2i·13-s − 2·14-s + 16-s + i·18-s + 4·19-s + 2·21-s − 6i·22-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + 0.577i·3-s − 0.5·4-s + 0.408·6-s − 0.755i·7-s + 0.353i·8-s − 0.333·9-s + 1.80·11-s − 0.288i·12-s + 0.554i·13-s − 0.534·14-s + 0.250·16-s + 0.235i·18-s + 0.917·19-s + 0.436·21-s − 1.27i·22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.980793586\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.980793586\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 23 | \( 1 - iT \) |
good | 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 - 6T + 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 + 4iT - 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - 8iT - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 - 2iT - 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 + 6iT - 73T^{2} \) |
| 79 | \( 1 - 14T + 79T^{2} \) |
| 83 | \( 1 - 6iT - 83T^{2} \) |
| 89 | \( 1 - 16T + 89T^{2} \) |
| 97 | \( 1 - 2iT - 97T^{2} \) |
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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
−8.993883241450666337816134473600, −7.84710628657557413165449286363, −7.06760333700770119196010240133, −6.27870619126034133523431091677, −5.33154834770127421508309228595, −4.35341327271708545848543598055, −3.89350702763081584189925373415, −3.18464026185754385506578020511, −1.83776409882844902189692535062, −0.897680252116099174221353964138,
0.856774625750197840468610163156, 1.94954751434690787556334214103, 3.20186522614200635346623325059, 3.99520512635021935508111741411, 5.12395001214266594974895246504, 5.76856955182553138314877180714, 6.46435941574037630586814168836, 7.10081621892784105469471836501, 7.79567380590498445860869388904, 8.720089909715866857283094634176