| 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
−10.23329464756720269519501952407, −9.760495732942717985819936591379, −8.609356843080483474480513949284, −7.899980267027830944401240189752, −6.73519482891855758349620952875, −5.40969598714047347836151956300, −5.01680085486630376879797109998, −3.12160359901020048538613695460, −2.26945513601220936872196404569, −0.74937687554549699097085119247,
2.10180563193725297307983955965, 3.55211255126426616751106338385, 4.78021947125316864590275242887, 5.68045820068518689822331327989, 6.46297204943467024401033728284, 7.47683564774457667421793465899, 8.565529098615976236460109950961, 9.288803807794320598797227422516, 10.34091002365277322260252363673, 10.62377838197558305350434494111
