L(s) = 1 | + (−0.309 − 0.951i)2-s + (0.309 − 0.951i)3-s + (−0.809 + 0.587i)4-s − 5-s − 0.999·6-s + (1.31 − 0.957i)7-s + (0.809 + 0.587i)8-s + (−0.809 − 0.587i)9-s + (0.309 + 0.951i)10-s + (−0.438 + 0.318i)11-s + (0.309 + 0.951i)12-s + (−1.26 + 3.90i)13-s + (−1.31 − 0.957i)14-s + (−0.309 + 0.951i)15-s + (0.309 − 0.951i)16-s + (−4.63 − 3.37i)17-s + ⋯ |
L(s) = 1 | + (−0.218 − 0.672i)2-s + (0.178 − 0.549i)3-s + (−0.404 + 0.293i)4-s − 0.447·5-s − 0.408·6-s + (0.498 − 0.361i)7-s + (0.286 + 0.207i)8-s + (−0.269 − 0.195i)9-s + (0.0977 + 0.300i)10-s + (−0.132 + 0.0960i)11-s + (0.0892 + 0.274i)12-s + (−0.352 + 1.08i)13-s + (−0.352 − 0.255i)14-s + (−0.0797 + 0.245i)15-s + (0.0772 − 0.237i)16-s + (−1.12 − 0.817i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.860 - 0.508i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.860 - 0.508i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.134246 + 0.490850i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.134246 + 0.490850i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.309 + 0.951i)T \) |
| 3 | \( 1 + (-0.309 + 0.951i)T \) |
| 5 | \( 1 + T \) |
| 31 | \( 1 + (5.47 + 1.00i)T \) |
good | 7 | \( 1 + (-1.31 + 0.957i)T + (2.16 - 6.65i)T^{2} \) |
| 11 | \( 1 + (0.438 - 0.318i)T + (3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (1.26 - 3.90i)T + (-10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (4.63 + 3.37i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (1.96 + 6.04i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (6.13 + 4.46i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (0.0513 + 0.157i)T + (-23.4 + 17.0i)T^{2} \) |
| 37 | \( 1 - 3.76T + 37T^{2} \) |
| 41 | \( 1 + (-2.66 - 8.21i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + (-0.369 - 1.13i)T + (-34.7 + 25.2i)T^{2} \) |
| 47 | \( 1 + (-0.679 + 2.09i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (11.3 + 8.26i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (1.83 - 5.64i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 - 10.3T + 61T^{2} \) |
| 67 | \( 1 + 7.49T + 67T^{2} \) |
| 71 | \( 1 + (8.09 + 5.88i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (5.10 - 3.70i)T + (22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (-4.55 - 3.30i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (3.17 + 9.77i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (-2.10 + 1.52i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (-15.1 + 10.9i)T + (29.9 - 92.2i)T^{2} \) |
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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
−9.447696556571305367698617458390, −8.849399878652639280398244061310, −7.945433732419868738118233018235, −7.17365717508945231091073623470, −6.37668833887165024661354781173, −4.71413965115890191557416807810, −4.26939428525260469587758897904, −2.76384205216231523444698089496, −1.87975261618902981261494191972, −0.23914802263668298805643916396,
1.97687841928968996172198760551, 3.55820118849861113518657743394, 4.38667013490751620307219444142, 5.48991748095802511020755297498, 6.08537945610770421992366826058, 7.46301203565521038947280715318, 8.083995282351769155003253034751, 8.673532082871124342458663519488, 9.647426851714502996876768813960, 10.47778871725895003369843975589