| L(s) = 1 | + i·2-s + i·3-s − 4-s + (2 + i)5-s − 6-s − i·8-s − 9-s + (−1 + 2i)10-s − 4·11-s − i·12-s + 4i·13-s + (−1 + 2i)15-s + 16-s + 6i·17-s − i·18-s + 19-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + 0.577i·3-s − 0.5·4-s + (0.894 + 0.447i)5-s − 0.408·6-s − 0.353i·8-s − 0.333·9-s + (−0.316 + 0.632i)10-s − 1.20·11-s − 0.288i·12-s + 1.10i·13-s + (−0.258 + 0.516i)15-s + 0.250·16-s + 1.45i·17-s − 0.235i·18-s + 0.229·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.304357 + 1.28928i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.304357 + 1.28928i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - iT \) |
| 5 | \( 1 + (-2 - i)T \) |
| 19 | \( 1 - T \) |
| good | 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 - 4iT - 13T^{2} \) |
| 17 | \( 1 - 6iT - 17T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 4iT - 37T^{2} \) |
| 41 | \( 1 + 12T + 41T^{2} \) |
| 43 | \( 1 + 6iT - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 - 14iT - 53T^{2} \) |
| 59 | \( 1 - 10T + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 + 8iT - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 - 8T + 89T^{2} \) |
| 97 | \( 1 + 10iT - 97T^{2} \) |
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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
−10.62377838197558305350434494111, −10.34091002365277322260252363673, −9.288803807794320598797227422516, −8.565529098615976236460109950961, −7.47683564774457667421793465899, −6.46297204943467024401033728284, −5.68045820068518689822331327989, −4.78021947125316864590275242887, −3.55211255126426616751106338385, −2.10180563193725297307983955965,
0.74937687554549699097085119247, 2.26945513601220936872196404569, 3.12160359901020048538613695460, 5.01680085486630376879797109998, 5.40969598714047347836151956300, 6.73519482891855758349620952875, 7.899980267027830944401240189752, 8.609356843080483474480513949284, 9.760495732942717985819936591379, 10.23329464756720269519501952407
