L(s) = 1 | + (−2.32 + 1.34i)2-s + (2.61 − 4.52i)4-s + (2.23 + 0.0881i)5-s + (3.14 + 1.81i)7-s + 8.65i·8-s + (−5.31 + 2.79i)10-s + (1.80 + 3.12i)11-s + (1.66 − 3.19i)13-s − 9.75·14-s + (−6.40 − 11.1i)16-s + (−1.37 − 0.795i)17-s + (−2.37 + 4.11i)19-s + (6.23 − 9.87i)20-s + (−8.40 − 4.85i)22-s + (3.63 − 2.10i)23-s + ⋯ |
L(s) = 1 | + (−1.64 + 0.950i)2-s + (1.30 − 2.26i)4-s + (0.999 + 0.0394i)5-s + (1.18 + 0.685i)7-s + 3.06i·8-s + (−1.68 + 0.884i)10-s + (0.544 + 0.943i)11-s + (0.461 − 0.887i)13-s − 2.60·14-s + (−1.60 − 2.77i)16-s + (−0.334 − 0.192i)17-s + (−0.544 + 0.943i)19-s + (1.39 − 2.20i)20-s + (−1.79 − 1.03i)22-s + (0.758 − 0.438i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.248 - 0.968i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.248 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.758143 + 0.588493i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.758143 + 0.588493i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (-2.23 - 0.0881i)T \) |
| 13 | \( 1 + (-1.66 + 3.19i)T \) |
good | 2 | \( 1 + (2.32 - 1.34i)T + (1 - 1.73i)T^{2} \) |
| 7 | \( 1 + (-3.14 - 1.81i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.80 - 3.12i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (1.37 + 0.795i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2.37 - 4.11i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.63 + 2.10i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (2.32 + 4.02i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 7.07T + 31T^{2} \) |
| 37 | \( 1 + (7.72 - 4.46i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (3.49 + 6.05i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (1.42 + 0.823i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 6.71iT - 47T^{2} \) |
| 53 | \( 1 - 6.29iT - 53T^{2} \) |
| 59 | \( 1 + (0.407 - 0.705i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (5.29 - 9.17i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-7.53 + 4.34i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (2.74 - 4.76i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 - 1.32iT - 73T^{2} \) |
| 79 | \( 1 + 6.22T + 79T^{2} \) |
| 83 | \( 1 - 6.63iT - 83T^{2} \) |
| 89 | \( 1 + (6.68 + 11.5i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-5.41 - 3.12i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.36306311945144986576775754321, −9.981812116177512511002905018471, −8.781935402749530090189127460762, −8.546476051976804949672688536495, −7.47049716280656938815370419458, −6.54051418007485324982334691742, −5.73026516074435817800740999766, −4.89977266198786546520346752880, −2.26402548745784558127492538001, −1.39720565312879788251973876468,
1.12981614246278472511841180300, 1.93658869544971362775585991348, 3.31526120813066332104574246922, 4.68246797081879727953609003932, 6.40903082132241753268157317267, 7.16580106345477621102848899999, 8.383859854164869313188954492702, 8.832106809562867811354189672408, 9.590144574167895204391574756474, 10.62644708196369647434891196642