L(s) = 1 | − 1.41i·2-s − 1.00·4-s + 1.41i·5-s − 7-s + 2.00·10-s − 11-s + 13-s + 1.41i·14-s − 0.999·16-s + 1.41i·19-s − 1.41i·20-s + 1.41i·22-s + 23-s − 1.00·25-s − 1.41i·26-s + ⋯ |
L(s) = 1 | − 1.41i·2-s − 1.00·4-s + 1.41i·5-s − 7-s + 2.00·10-s − 11-s + 13-s + 1.41i·14-s − 0.999·16-s + 1.41i·19-s − 1.41i·20-s + 1.41i·22-s + 23-s − 1.00·25-s − 1.41i·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2547 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2547 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8878379904\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8878379904\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 283 | \( 1 - T \) |
good | 2 | \( 1 + 1.41iT - T^{2} \) |
| 5 | \( 1 - 1.41iT - T^{2} \) |
| 7 | \( 1 + T + T^{2} \) |
| 11 | \( 1 + T + T^{2} \) |
| 13 | \( 1 - T + T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - 1.41iT - T^{2} \) |
| 23 | \( 1 - T + T^{2} \) |
| 29 | \( 1 - T + T^{2} \) |
| 31 | \( 1 - 1.41iT - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + T + T^{2} \) |
| 43 | \( 1 - 1.41iT - T^{2} \) |
| 47 | \( 1 - 1.41iT - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + T + T^{2} \) |
| 61 | \( 1 - T + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - 2T + T^{2} \) |
| 89 | \( 1 - T + T^{2} \) |
| 97 | \( 1 - T + T^{2} \) |
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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
−9.476172523956910093300225598684, −8.498360470112806885640681701212, −7.57372210931640267196474545457, −6.57307124950493755579005109479, −6.26354662231905343526706012635, −4.95493129489901418364889277747, −3.71205048834642910215329315534, −3.14109427540077014989611945302, −2.71962470473180361004305161461, −1.42175357850838866047713887836,
0.60362629309525608176123555523, 2.40054227820248313807731092192, 3.66083185311367021256757540631, 4.86955564260269628770000945605, 5.15097376811616061839467123194, 6.09044998557728189503409538270, 6.72246423115098863736403380844, 7.52258329585208277744141463931, 8.381786734725076970583318153397, 8.813031322149062031736096157151