| L(s) = 1 | + 2i·2-s − 3.06i·3-s − 4·4-s + (−9.26 + 6.25i)5-s + 6.13·6-s − 8i·8-s + 17.5·9-s + (−12.5 − 18.5i)10-s − 49.5·11-s + 12.2i·12-s + 44.6i·13-s + (19.1 + 28.4i)15-s + 16·16-s + 39.2i·17-s + 35.1i·18-s + 56.3·19-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.590i·3-s − 0.5·4-s + (−0.828 + 0.559i)5-s + 0.417·6-s − 0.353i·8-s + 0.651·9-s + (−0.395 − 0.586i)10-s − 1.35·11-s + 0.295i·12-s + 0.951i·13-s + (0.330 + 0.489i)15-s + 0.250·16-s + 0.560i·17-s + 0.460i·18-s + 0.679·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.559 + 0.828i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.559 + 0.828i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.8465356157\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8465356157\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2iT \) |
| 5 | \( 1 + (9.26 - 6.25i)T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 3.06iT - 27T^{2} \) |
| 11 | \( 1 + 49.5T + 1.33e3T^{2} \) |
| 13 | \( 1 - 44.6iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 39.2iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 56.3T + 6.85e3T^{2} \) |
| 23 | \( 1 + 91.6iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 281.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 70.5T + 2.97e4T^{2} \) |
| 37 | \( 1 + 197. iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 399.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 203. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 68.4iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 617. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 480.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 23.4T + 2.26e5T^{2} \) |
| 67 | \( 1 + 252. iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 835.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 257. iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 773.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.34e3iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 1.30e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 323. iT - 9.12e5T^{2} \) |
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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
−10.45052149652951540272879786629, −9.408060949310052239723193223488, −8.216244341644569246479157793434, −7.54119270136517808433666964418, −6.98052636420314384180840772377, −5.95185441031615626656082698081, −4.69311213924208609574951801735, −3.68319711735364527238677159171, −2.15788193258017018325941984086, −0.31734705134313680755478800226,
1.06225241110493665755751759558, 2.84037008159359265905638400018, 3.81904394117155721023404531737, 4.85410268307847119442407064540, 5.51671413781492368195813691591, 7.51995681351434362139439306445, 7.890287937728597666052671144637, 9.207989216625375152801395302364, 9.820716368105532291061731866930, 10.78838243312736971088016959663
