| L(s) = 1 | − 0.722·3-s + 0.383i·5-s − i·7-s − 2.47·9-s + i·11-s + (3.37 + 1.27i)13-s − 0.277i·15-s − 5.43·17-s + 5.21i·19-s + 0.722i·21-s + 2.81·23-s + 4.85·25-s + 3.95·27-s + 1.99·29-s − 2.19i·31-s + ⋯ |
| L(s) = 1 | − 0.417·3-s + 0.171i·5-s − 0.377i·7-s − 0.825·9-s + 0.301i·11-s + (0.935 + 0.353i)13-s − 0.0716i·15-s − 1.31·17-s + 1.19i·19-s + 0.157i·21-s + 0.586·23-s + 0.970·25-s + 0.762·27-s + 0.371·29-s − 0.393i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.935 - 0.353i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.935 - 0.353i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3649555969\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3649555969\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + iT \) |
| 11 | \( 1 - iT \) |
| 13 | \( 1 + (-3.37 - 1.27i)T \) |
| good | 3 | \( 1 + 0.722T + 3T^{2} \) |
| 5 | \( 1 - 0.383iT - 5T^{2} \) |
| 17 | \( 1 + 5.43T + 17T^{2} \) |
| 19 | \( 1 - 5.21iT - 19T^{2} \) |
| 23 | \( 1 - 2.81T + 23T^{2} \) |
| 29 | \( 1 - 1.99T + 29T^{2} \) |
| 31 | \( 1 + 2.19iT - 31T^{2} \) |
| 37 | \( 1 + 8.80iT - 37T^{2} \) |
| 41 | \( 1 - 5.89iT - 41T^{2} \) |
| 43 | \( 1 + 9.25T + 43T^{2} \) |
| 47 | \( 1 + 8.28iT - 47T^{2} \) |
| 53 | \( 1 + 0.632T + 53T^{2} \) |
| 59 | \( 1 + 13.2iT - 59T^{2} \) |
| 61 | \( 1 + 13.3T + 61T^{2} \) |
| 67 | \( 1 - 7.84iT - 67T^{2} \) |
| 71 | \( 1 - 8.46iT - 71T^{2} \) |
| 73 | \( 1 - 13.6iT - 73T^{2} \) |
| 79 | \( 1 + 13.9T + 79T^{2} \) |
| 83 | \( 1 - 13.5iT - 83T^{2} \) |
| 89 | \( 1 + 3.98iT - 89T^{2} \) |
| 97 | \( 1 - 4.98iT - 97T^{2} \) |
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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
−8.597995051266613570246801705653, −8.255516359732080993290358237266, −7.08964567020634499086289340168, −6.59119506329760776719840156602, −5.88868660466297385265948925081, −5.07467070864666516989202872356, −4.22436154600892869061912213417, −3.42847468574235333079756777229, −2.41328391846593415836975827513, −1.28625940444814624025624080893,
0.11785076940784983380772440067, 1.34426890691813442569978122299, 2.73003856605130049876459418329, 3.23332387270482289596245472481, 4.64242403253109417187194679710, 4.99161945005139957304478631578, 6.08177513365083431511564132984, 6.41960094764201219956634566147, 7.30319263290538897323395706572, 8.436875438087322116113964880652
