| L(s) = 1 | + i·2-s + 3-s − 4-s − 3i·5-s + i·6-s + i·7-s − i·8-s + 9-s + 3·10-s − 5i·11-s − 12-s + (−3 − 2i)13-s − 14-s − 3i·15-s + 16-s + 3·17-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + 0.577·3-s − 0.5·4-s − 1.34i·5-s + 0.408i·6-s + 0.377i·7-s − 0.353i·8-s + 0.333·9-s + 0.948·10-s − 1.50i·11-s − 0.288·12-s + (−0.832 − 0.554i)13-s − 0.267·14-s − 0.774i·15-s + 0.250·16-s + 0.727·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 + 0.554i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.832 + 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.47541 - 0.446720i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.47541 - 0.446720i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 - iT \) |
| 13 | \( 1 + (3 + 2i)T \) |
| good | 5 | \( 1 + 3iT - 5T^{2} \) |
| 11 | \( 1 + 5iT - 11T^{2} \) |
| 17 | \( 1 - 3T + 17T^{2} \) |
| 19 | \( 1 + iT - 19T^{2} \) |
| 23 | \( 1 + T + 23T^{2} \) |
| 29 | \( 1 - 5T + 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 + 7iT - 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + T + 43T^{2} \) |
| 47 | \( 1 - 8iT - 47T^{2} \) |
| 53 | \( 1 - 14T + 53T^{2} \) |
| 59 | \( 1 - 14iT - 59T^{2} \) |
| 61 | \( 1 + 3T + 61T^{2} \) |
| 67 | \( 1 - 8iT - 67T^{2} \) |
| 71 | \( 1 + 10iT - 71T^{2} \) |
| 73 | \( 1 - 11iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 6iT - 83T^{2} \) |
| 89 | \( 1 + 16iT - 89T^{2} \) |
| 97 | \( 1 + 2iT - 97T^{2} \) |
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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
−10.48578500125122597595345153564, −9.474835345348604022768159047682, −8.717431507189508441762219730357, −8.253158637349628081886891555853, −7.33807092747286015892275102407, −5.89706123978545200226975534533, −5.28186697409185948060013263027, −4.18675780044876769409520304301, −2.86349550330828889337040673671, −0.881998858320457016092041279158,
1.92143899985575659444802638883, 2.86521533116791941437839235681, 3.92534040678059442665584062182, 4.96054312587536855532952426758, 6.65867049724933434258730447769, 7.26015157601889847259167581047, 8.179094225529055222791246943536, 9.566490198455084588934479518599, 10.03045588248703020413097068827, 10.64424565668071911071373429551
