L(s) = 1 | + (−0.809 − 0.587i)2-s + (−0.809 + 0.587i)3-s + (0.309 + 0.951i)4-s − 5-s + 6-s + (0.0931 + 0.286i)7-s + (0.309 − 0.951i)8-s + (0.309 − 0.951i)9-s + (0.809 + 0.587i)10-s + (−0.593 − 1.82i)11-s + (−0.809 − 0.587i)12-s + (1.37 − 0.997i)13-s + (0.0931 − 0.286i)14-s + (0.809 − 0.587i)15-s + (−0.809 + 0.587i)16-s + (−1.87 + 5.76i)17-s + ⋯ |
L(s) = 1 | + (−0.572 − 0.415i)2-s + (−0.467 + 0.339i)3-s + (0.154 + 0.475i)4-s − 0.447·5-s + 0.408·6-s + (0.0352 + 0.108i)7-s + (0.109 − 0.336i)8-s + (0.103 − 0.317i)9-s + (0.255 + 0.185i)10-s + (−0.178 − 0.550i)11-s + (−0.233 − 0.169i)12-s + (0.380 − 0.276i)13-s + (0.0248 − 0.0766i)14-s + (0.208 − 0.151i)15-s + (−0.202 + 0.146i)16-s + (−0.454 + 1.39i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0388 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0388 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.437292 - 0.454634i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.437292 - 0.454634i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.809 + 0.587i)T \) |
| 3 | \( 1 + (0.809 - 0.587i)T \) |
| 5 | \( 1 + T \) |
| 31 | \( 1 + (-3.38 + 4.42i)T \) |
good | 7 | \( 1 + (-0.0931 - 0.286i)T + (-5.66 + 4.11i)T^{2} \) |
| 11 | \( 1 + (0.593 + 1.82i)T + (-8.89 + 6.46i)T^{2} \) |
| 13 | \( 1 + (-1.37 + 0.997i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (1.87 - 5.76i)T + (-13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (0.820 + 0.596i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (-0.226 + 0.697i)T + (-18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (6.77 + 4.92i)T + (8.96 + 27.5i)T^{2} \) |
| 37 | \( 1 - 5.63T + 37T^{2} \) |
| 41 | \( 1 + (8.62 + 6.26i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + (-3.50 - 2.54i)T + (13.2 + 40.8i)T^{2} \) |
| 47 | \( 1 + (-6.97 + 5.06i)T + (14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (-3.79 + 11.6i)T + (-42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (-10.0 + 7.26i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + 4.04T + 61T^{2} \) |
| 67 | \( 1 + 8.33T + 67T^{2} \) |
| 71 | \( 1 + (4.12 - 12.6i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (4.20 + 12.9i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (-4.36 + 13.4i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (4.46 + 3.24i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + (1.55 + 4.77i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (-3.23 - 9.96i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−10.08614124472866369414288556973, −8.949711657874704362443307843115, −8.376730130793715927904784546710, −7.50529751964003268128369883196, −6.37326237045126323862811283761, −5.59359686336464673839614665403, −4.24675432734222070435809698859, −3.54073473293981800525333172408, −2.11041143285199463116146997403, −0.43438556903692472001332084362,
1.18490246431982434489720430461, 2.66397034949611589734120704719, 4.24191388980049273661535648316, 5.15217339285353284725208776705, 6.14439386053449430800992976829, 7.17890390421823974869753574400, 7.45584068027279720031261144209, 8.632892378406374664047682897839, 9.310174107279513366210872749128, 10.29980133826237412294972477430