| L(s) = 1 | + 2·4-s + i·7-s + 3·11-s − 5i·13-s + 4·16-s − 3i·17-s − 2·19-s − 6i·23-s + 2i·28-s + 3·29-s − 4·31-s + 2i·37-s + 12·41-s + 10i·43-s + 6·44-s + ⋯ |
| L(s) = 1 | + 4-s + 0.377i·7-s + 0.904·11-s − 1.38i·13-s + 16-s − 0.727i·17-s − 0.458·19-s − 1.25i·23-s + 0.377i·28-s + 0.557·29-s − 0.718·31-s + 0.328i·37-s + 1.87·41-s + 1.52i·43-s + 0.904·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.310002049\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.310002049\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - iT \) |
| good | 2 | \( 1 - 2T^{2} \) |
| 11 | \( 1 - 3T + 11T^{2} \) |
| 13 | \( 1 + 5iT - 13T^{2} \) |
| 17 | \( 1 + 3iT - 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 + 6iT - 23T^{2} \) |
| 29 | \( 1 - 3T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 - 12T + 41T^{2} \) |
| 43 | \( 1 - 10iT - 43T^{2} \) |
| 47 | \( 1 + 9iT - 47T^{2} \) |
| 53 | \( 1 - 12iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 8T + 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 2iT - 73T^{2} \) |
| 79 | \( 1 - T + 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 + 12T + 89T^{2} \) |
| 97 | \( 1 + iT - 97T^{2} \) |
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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
−9.399349594446858084312927631376, −8.476032780663399742993589289873, −7.75858576815090926024565943630, −6.88226149251641680396258570888, −6.18613744789754897364081728834, −5.44082946768127261295685161203, −4.28902274945067524053534047938, −3.08984098876031870664575516273, −2.38979472144677827315004858676, −0.984811152605994644250568524101,
1.37921665283021032989857998998, 2.22062672683477142719393612124, 3.60758126763942390625852856145, 4.22280382744029398440466937073, 5.58416559008108437440761062076, 6.40995117947110185873739957064, 6.99040980534900117286456856642, 7.71096125778843593527190079586, 8.752835688899261381175208934640, 9.478365874869076816654096198730
