L(s) = 1 | + i·3-s + 7-s − 9-s − 2i·11-s + 2i·13-s + 4·17-s + 4i·19-s + i·21-s + 6·23-s + 5·25-s − i·27-s + 2i·29-s + 2·33-s − 6i·37-s − 2·39-s + ⋯ |
L(s) = 1 | + 0.577i·3-s + 0.377·7-s − 0.333·9-s − 0.603i·11-s + 0.554i·13-s + 0.970·17-s + 0.917i·19-s + 0.218i·21-s + 1.25·23-s + 25-s − 0.192i·27-s + 0.371i·29-s + 0.348·33-s − 0.986i·37-s − 0.320·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5376 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5376 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.160376678\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.160376678\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 7 | \( 1 - T \) |
good | 5 | \( 1 - 5T^{2} \) |
| 11 | \( 1 + 2iT - 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 - 4T + 17T^{2} \) |
| 19 | \( 1 - 4iT - 19T^{2} \) |
| 23 | \( 1 - 6T + 23T^{2} \) |
| 29 | \( 1 - 2iT - 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 6iT - 37T^{2} \) |
| 41 | \( 1 + 8T + 41T^{2} \) |
| 43 | \( 1 + 8iT - 43T^{2} \) |
| 47 | \( 1 + 4T + 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 - 59T^{2} \) |
| 61 | \( 1 - 14iT - 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 - 2T + 71T^{2} \) |
| 73 | \( 1 - 2T + 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 6T + 97T^{2} \) |
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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
−8.482407267518918177009728247817, −7.55090942509323798610577960848, −6.89312458875876369594453513536, −5.98187034774515033539097406451, −5.28305722586905312944351402627, −4.71513313128057325711907366735, −3.66067661906793563971256821621, −3.20410245219993126917329645757, −1.99630062897520423637230473211, −0.894927630338655396370317804238,
0.76256401842454415998749725202, 1.63787614739273291174689575915, 2.77533863553591508732474637049, 3.33530731953269324470712018333, 4.72563675381539382616282028767, 5.02118610371483184584900692641, 6.00147689532617702060134140399, 6.83018988539595611799367683839, 7.28849655026615060729648131889, 8.110270550611922127619278200764