L(s) = 1 | + (−0.707 − 0.707i)2-s + (1.33 + 1.10i)3-s + 1.00i·4-s + (0.707 − 0.707i)5-s + (−0.159 − 1.72i)6-s − 3.17·7-s + (0.707 − 0.707i)8-s + (0.549 + 2.94i)9-s − 1.00·10-s + 4.29·11-s + (−1.10 + 1.33i)12-s + (0.469 + 0.469i)13-s + (2.24 + 2.24i)14-s + (1.72 − 0.159i)15-s − 1.00·16-s + (−4.17 + 4.17i)17-s + ⋯ |
L(s) = 1 | + (−0.499 − 0.499i)2-s + (0.769 + 0.639i)3-s + 0.500i·4-s + (0.316 − 0.316i)5-s + (−0.0650 − 0.704i)6-s − 1.19·7-s + (0.250 − 0.250i)8-s + (0.183 + 0.983i)9-s − 0.316·10-s + 1.29·11-s + (−0.319 + 0.384i)12-s + (0.130 + 0.130i)13-s + (0.599 + 0.599i)14-s + (0.445 − 0.0411i)15-s − 0.250·16-s + (−1.01 + 1.01i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.545 - 0.838i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.545 - 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.464231431\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.464231431\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 + 0.707i)T \) |
| 3 | \( 1 + (-1.33 - 1.10i)T \) |
| 5 | \( 1 + (-0.707 + 0.707i)T \) |
| 37 | \( 1 + (-3.36 + 5.06i)T \) |
good | 7 | \( 1 + 3.17T + 7T^{2} \) |
| 11 | \( 1 - 4.29T + 11T^{2} \) |
| 13 | \( 1 + (-0.469 - 0.469i)T + 13iT^{2} \) |
| 17 | \( 1 + (4.17 - 4.17i)T - 17iT^{2} \) |
| 19 | \( 1 + (-2.35 - 2.35i)T + 19iT^{2} \) |
| 23 | \( 1 + (2.43 - 2.43i)T - 23iT^{2} \) |
| 29 | \( 1 + (-3.28 - 3.28i)T + 29iT^{2} \) |
| 31 | \( 1 + (0.0756 - 0.0756i)T - 31iT^{2} \) |
| 41 | \( 1 - 6.28T + 41T^{2} \) |
| 43 | \( 1 + (-2.94 - 2.94i)T + 43iT^{2} \) |
| 47 | \( 1 - 0.554iT - 47T^{2} \) |
| 53 | \( 1 - 10.4iT - 53T^{2} \) |
| 59 | \( 1 + (-5.42 + 5.42i)T - 59iT^{2} \) |
| 61 | \( 1 + (3.80 - 3.80i)T - 61iT^{2} \) |
| 67 | \( 1 - 7.22iT - 67T^{2} \) |
| 71 | \( 1 - 8.00iT - 71T^{2} \) |
| 73 | \( 1 + 12.0iT - 73T^{2} \) |
| 79 | \( 1 + (2.36 + 2.36i)T + 79iT^{2} \) |
| 83 | \( 1 - 10.3iT - 83T^{2} \) |
| 89 | \( 1 + (-2.62 - 2.62i)T + 89iT^{2} \) |
| 97 | \( 1 + (0.130 + 0.130i)T + 97iT^{2} \) |
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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
−9.767026877367524326121157423513, −9.223255513274989797546179467183, −8.758429717361086150180640519467, −7.74725712058068532792734119888, −6.67692873139692238944924809620, −5.82742982742705327040269337909, −4.27581671695999378085161524626, −3.75245802732918922893287483342, −2.69537978192420827742575129163, −1.48523313045275244095535275236,
0.72849740035425698427832363658, 2.26581832549843367568132288144, 3.21756063915729940801057934245, 4.38045162554329288022718950799, 5.96629990930810180734537828847, 6.66587738978822742784921957266, 6.98908418020245109875519014947, 8.084450652629747755683695680662, 9.055932332526140668259775192721, 9.417198914422842449258506044020