L(s) = 1 | + 3i·3-s + i·7-s − 6·9-s + 5·11-s + 3i·13-s − i·17-s + 6·19-s − 3·21-s + 6i·23-s − 9i·27-s + 9·29-s + 4·31-s + 15i·33-s + 2i·37-s − 9·39-s + ⋯ |
L(s) = 1 | + 1.73i·3-s + 0.377i·7-s − 2·9-s + 1.50·11-s + 0.832i·13-s − 0.242i·17-s + 1.37·19-s − 0.654·21-s + 1.25i·23-s − 1.73i·27-s + 1.67·29-s + 0.718·31-s + 2.61i·33-s + 0.328i·37-s − 1.44·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.967583873\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.967583873\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - iT \) |
good | 3 | \( 1 - 3iT - 3T^{2} \) |
| 11 | \( 1 - 5T + 11T^{2} \) |
| 13 | \( 1 - 3iT - 13T^{2} \) |
| 17 | \( 1 + iT - 17T^{2} \) |
| 19 | \( 1 - 6T + 19T^{2} \) |
| 23 | \( 1 - 6iT - 23T^{2} \) |
| 29 | \( 1 - 9T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 + 4T + 41T^{2} \) |
| 43 | \( 1 - 10iT - 43T^{2} \) |
| 47 | \( 1 - iT - 47T^{2} \) |
| 53 | \( 1 + 4iT - 53T^{2} \) |
| 59 | \( 1 + 8T + 59T^{2} \) |
| 61 | \( 1 + 8T + 61T^{2} \) |
| 67 | \( 1 + 12iT - 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 + 2iT - 73T^{2} \) |
| 79 | \( 1 - 13T + 79T^{2} \) |
| 83 | \( 1 + 4iT - 83T^{2} \) |
| 89 | \( 1 + 4T + 89T^{2} \) |
| 97 | \( 1 + 13iT - 97T^{2} \) |
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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
−9.398174719608895208671737881164, −8.723141604878883079882675332289, −7.78065667224457195995116578000, −6.62140291844918863942023604177, −5.98924522640846236766681601453, −4.92433557260171185480729950246, −4.53576490497801236392754977217, −3.54897168076766094487351663615, −3.00032808649811991417742536225, −1.40040218579437681303907033798,
0.73741687274224139332583221697, 1.35112956324768647266050146326, 2.55240628993546176703583748691, 3.41204131006809852113083302784, 4.58725350054203323944738814249, 5.70750324547353301729639771179, 6.42213347772441775879381672322, 6.93309538377953693684717319432, 7.60715857593992724692051154627, 8.371968549762333853553724799979