| L(s) = 1 | + 1.33i·3-s − 2.57i·5-s + (−2.48 + 0.918i)7-s + 1.21·9-s + (3.23 − 0.742i)11-s − 13-s + 3.44·15-s − 4.02·17-s − 2.61·19-s + (−1.22 − 3.31i)21-s − 3.29·23-s − 1.63·25-s + 5.63i·27-s − 1.57i·29-s + 6.91i·31-s + ⋯ |
| L(s) = 1 | + 0.771i·3-s − 1.15i·5-s + (−0.937 + 0.347i)7-s + 0.404·9-s + (0.974 − 0.223i)11-s − 0.277·13-s + 0.888·15-s − 0.976·17-s − 0.599·19-s + (−0.267 − 0.723i)21-s − 0.686·23-s − 0.327·25-s + 1.08i·27-s − 0.292i·29-s + 1.24i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.836 - 0.548i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.836 - 0.548i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.637189162\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.637189162\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (2.48 - 0.918i)T \) |
| 11 | \( 1 + (-3.23 + 0.742i)T \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 - 1.33iT - 3T^{2} \) |
| 5 | \( 1 + 2.57iT - 5T^{2} \) |
| 17 | \( 1 + 4.02T + 17T^{2} \) |
| 19 | \( 1 + 2.61T + 19T^{2} \) |
| 23 | \( 1 + 3.29T + 23T^{2} \) |
| 29 | \( 1 + 1.57iT - 29T^{2} \) |
| 31 | \( 1 - 6.91iT - 31T^{2} \) |
| 37 | \( 1 - 7.23T + 37T^{2} \) |
| 41 | \( 1 - 4.62T + 41T^{2} \) |
| 43 | \( 1 - 3.28iT - 43T^{2} \) |
| 47 | \( 1 - 11.0iT - 47T^{2} \) |
| 53 | \( 1 - 10.3T + 53T^{2} \) |
| 59 | \( 1 + 11.2iT - 59T^{2} \) |
| 61 | \( 1 - 11.5T + 61T^{2} \) |
| 67 | \( 1 + 11.9T + 67T^{2} \) |
| 71 | \( 1 - 0.0198T + 71T^{2} \) |
| 73 | \( 1 - 13.4T + 73T^{2} \) |
| 79 | \( 1 + 8.42iT - 79T^{2} \) |
| 83 | \( 1 - 5.30T + 83T^{2} \) |
| 89 | \( 1 + 12.3iT - 89T^{2} \) |
| 97 | \( 1 - 10.1iT - 97T^{2} \) |
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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
−8.818241980636107201169274115343, −7.981571948155175656886305849239, −6.87003837535086880440085601443, −6.30269413194291483506036300109, −5.45891077054686998505400505144, −4.48443317235366730108045511977, −4.21312983285169133005757995942, −3.22856336734220942605284813405, −2.04015218385284087665385357702, −0.819020370925297076364297105874,
0.64842845053217883413979251973, 2.06360983809375078824185878056, 2.64343618996854676512757529874, 3.90726201956721439535507421364, 4.20720227465978809629463140435, 5.77952001470005583255555022731, 6.47068906540300494260185650576, 6.91357931257090480330218195185, 7.29587413417185175874193782803, 8.240393078852741179406462403422
