| L(s) = 1 | + 1.55·3-s + 2.57i·5-s + i·7-s − 0.586·9-s + i·11-s + (−3.41 + 1.15i)13-s + 3.99i·15-s + 1.32·17-s − 6.23i·19-s + 1.55i·21-s − 5.37·23-s − 1.62·25-s − 5.57·27-s − 4.26·29-s − 0.0711i·31-s + ⋯ |
| L(s) = 1 | + 0.896·3-s + 1.15i·5-s + 0.377i·7-s − 0.195·9-s + 0.301i·11-s + (−0.947 + 0.321i)13-s + 1.03i·15-s + 0.320·17-s − 1.42i·19-s + 0.339i·21-s − 1.12·23-s − 0.325·25-s − 1.07·27-s − 0.791·29-s − 0.0127i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.947 + 0.321i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.947 + 0.321i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5369670002\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5369670002\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (3.41 - 1.15i)T \) |
| good | 3 | \( 1 - 1.55T + 3T^{2} \) |
| 5 | \( 1 - 2.57iT - 5T^{2} \) |
| 17 | \( 1 - 1.32T + 17T^{2} \) |
| 19 | \( 1 + 6.23iT - 19T^{2} \) |
| 23 | \( 1 + 5.37T + 23T^{2} \) |
| 29 | \( 1 + 4.26T + 29T^{2} \) |
| 31 | \( 1 + 0.0711iT - 31T^{2} \) |
| 37 | \( 1 + 8.75iT - 37T^{2} \) |
| 41 | \( 1 - 10.4iT - 41T^{2} \) |
| 43 | \( 1 + 6.31T + 43T^{2} \) |
| 47 | \( 1 - 10.7iT - 47T^{2} \) |
| 53 | \( 1 + 2.61T + 53T^{2} \) |
| 59 | \( 1 - 1.71iT - 59T^{2} \) |
| 61 | \( 1 - 2.48T + 61T^{2} \) |
| 67 | \( 1 - 4.22iT - 67T^{2} \) |
| 71 | \( 1 + 8.91iT - 71T^{2} \) |
| 73 | \( 1 + 4.27iT - 73T^{2} \) |
| 79 | \( 1 - 10.7T + 79T^{2} \) |
| 83 | \( 1 - 4.38iT - 83T^{2} \) |
| 89 | \( 1 + 10.8iT - 89T^{2} \) |
| 97 | \( 1 + 18.8iT - 97T^{2} \) |
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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
−8.961158529290662821961371759627, −7.956799574125218128201385319497, −7.52292353607945509346493531049, −6.76357709374361897189208341492, −6.04493084256001470322164821274, −5.08611423332083527637178696159, −4.15762806583845700652265952183, −3.15634846075226357080234059691, −2.64056749216649961984014467702, −1.94218013368871452152927544870,
0.12124953231357472595021404802, 1.48261786134300045237885733512, 2.36956947249184520570607551693, 3.51211018628991249229430071990, 4.00238513214861427257422566183, 5.16690228228367328524366386197, 5.55542069546243484840939604953, 6.64791778234481175254225697542, 7.67407314973660432856166955503, 8.151377800392421550057789346971
