| L(s) = 1 | + 2-s − i·3-s + 4-s − i·6-s + 2i·7-s + 8-s + 2·9-s − i·12-s − 13-s + 2i·14-s + 16-s + (4 + i)17-s + 2·18-s + 5·19-s + 2·21-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577i·3-s + 0.5·4-s − 0.408i·6-s + 0.755i·7-s + 0.353·8-s + 0.666·9-s − 0.288i·12-s − 0.277·13-s + 0.534i·14-s + 0.250·16-s + (0.970 + 0.242i)17-s + 0.471·18-s + 1.14·19-s + 0.436·21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 + 0.242i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.970 + 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.60261 - 0.320396i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.60261 - 0.320396i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 \) |
| 17 | \( 1 + (-4 - i)T \) |
| good | 3 | \( 1 + iT - 3T^{2} \) |
| 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + T + 13T^{2} \) |
| 19 | \( 1 - 5T + 19T^{2} \) |
| 23 | \( 1 - 4iT - 23T^{2} \) |
| 29 | \( 1 + 9iT - 29T^{2} \) |
| 31 | \( 1 + 5iT - 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 - 10iT - 41T^{2} \) |
| 43 | \( 1 + 6T + 43T^{2} \) |
| 47 | \( 1 + 7T + 47T^{2} \) |
| 53 | \( 1 + T + 53T^{2} \) |
| 59 | \( 1 - 5T + 59T^{2} \) |
| 61 | \( 1 + 5iT - 61T^{2} \) |
| 67 | \( 1 + 2T + 67T^{2} \) |
| 71 | \( 1 + 5iT - 71T^{2} \) |
| 73 | \( 1 + 11iT - 73T^{2} \) |
| 79 | \( 1 - 16iT - 79T^{2} \) |
| 83 | \( 1 + 6T + 83T^{2} \) |
| 89 | \( 1 + 5T + 89T^{2} \) |
| 97 | \( 1 - 7iT - 97T^{2} \) |
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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
−9.948641595155622724906423668754, −9.606726377162232337801964473565, −8.077919306531121782607033610071, −7.62183018120710943924401387811, −6.56396981228741290021739999661, −5.77246566520874668643280087176, −4.90599867604073383561679441293, −3.72408490421509416741385823229, −2.60317165587054514107490114548, −1.40528697224639712957051368926,
1.34861656048393514611469361948, 3.08345380489578168800755838317, 3.86995478855093629380648526161, 4.85446659007902882617174538775, 5.50672909255093926138028563137, 6.99279709010121128125677233959, 7.26198900944624484318785370612, 8.536997834117582741422688249575, 9.660935386854988744546368961587, 10.30308022931034358302261466968
