L(s) = 1 | + (0.258 + 0.965i)2-s + (−0.866 + 0.499i)4-s + (−0.866 − 0.5i)7-s + (−0.707 − 0.707i)8-s + (0.707 − 1.22i)11-s + (0.258 − 0.965i)14-s + (0.500 − 0.866i)16-s + (1.36 + 0.366i)22-s + (−0.707 − 1.22i)23-s + (0.5 − 0.866i)25-s + 28-s + (−1.22 − 0.707i)29-s + (0.965 + 0.258i)32-s + (1.73 + i)43-s + 1.41i·44-s + ⋯ |
L(s) = 1 | + (0.258 + 0.965i)2-s + (−0.866 + 0.499i)4-s + (−0.866 − 0.5i)7-s + (−0.707 − 0.707i)8-s + (0.707 − 1.22i)11-s + (0.258 − 0.965i)14-s + (0.500 − 0.866i)16-s + (1.36 + 0.366i)22-s + (−0.707 − 1.22i)23-s + (0.5 − 0.866i)25-s + 28-s + (−1.22 − 0.707i)29-s + (0.965 + 0.258i)32-s + (1.73 + i)43-s + 1.41i·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9496657672\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9496657672\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.258 - 0.965i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (0.866 + 0.5i)T \) |
good | 5 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (-1.73 - i)T + (0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + 1.41iT - T^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 - 1.41T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + (0.5 + 0.866i)T^{2} \) |
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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
−9.070341946315459236212681389827, −8.303814857402210069994147090680, −7.63463388909296287815197684203, −6.54228525660055638702034479279, −6.33503788246383559073533488251, −5.45495108854670781980364888589, −4.25115582369342345356476355430, −3.75202307320419476176695366681, −2.71981026190877422103082971316, −0.62927965551947854831800183477,
1.50288298545710665236787394470, 2.43630645492388514852437596002, 3.51586719957456649088036185138, 4.11717335814216680701547779325, 5.24155238284458194913831408418, 5.85869756228590605182305750979, 6.88556934824235437045714777132, 7.66793951444336030980712778426, 8.979125982762951274729402531575, 9.306535901529291348844638546022