L(s) = 1 | + (−0.809 − 0.587i)2-s + (−0.104 − 0.994i)3-s + (0.309 + 0.951i)4-s + (−0.5 − 0.866i)5-s + (−0.5 + 0.866i)6-s + (−0.813 − 0.172i)7-s + (0.309 − 0.951i)8-s + (−0.978 + 0.207i)9-s + (−0.104 + 0.994i)10-s + (2.87 − 3.19i)11-s + (0.913 − 0.406i)12-s + (5.10 + 2.27i)13-s + (0.556 + 0.617i)14-s + (−0.809 + 0.587i)15-s + (−0.809 + 0.587i)16-s + (1.58 + 1.75i)17-s + ⋯ |
L(s) = 1 | + (−0.572 − 0.415i)2-s + (−0.0603 − 0.574i)3-s + (0.154 + 0.475i)4-s + (−0.223 − 0.387i)5-s + (−0.204 + 0.353i)6-s + (−0.307 − 0.0653i)7-s + (0.109 − 0.336i)8-s + (−0.326 + 0.0693i)9-s + (−0.0330 + 0.314i)10-s + (0.866 − 0.961i)11-s + (0.263 − 0.117i)12-s + (1.41 + 0.630i)13-s + (0.148 + 0.165i)14-s + (−0.208 + 0.151i)15-s + (−0.202 + 0.146i)16-s + (0.384 + 0.426i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.470 + 0.882i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.470 + 0.882i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.556437 - 0.927057i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.556437 - 0.927057i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.104 + 0.994i)T \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
| 31 | \( 1 + (2.94 + 4.72i)T \) |
good | 7 | \( 1 + (0.813 + 0.172i)T + (6.39 + 2.84i)T^{2} \) |
| 11 | \( 1 + (-2.87 + 3.19i)T + (-1.14 - 10.9i)T^{2} \) |
| 13 | \( 1 + (-5.10 - 2.27i)T + (8.69 + 9.66i)T^{2} \) |
| 17 | \( 1 + (-1.58 - 1.75i)T + (-1.77 + 16.9i)T^{2} \) |
| 19 | \( 1 + (-1.60 + 0.716i)T + (12.7 - 14.1i)T^{2} \) |
| 23 | \( 1 + (-0.945 + 2.91i)T + (-18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (4.43 + 3.22i)T + (8.96 + 27.5i)T^{2} \) |
| 37 | \( 1 + (2.33 - 4.04i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.770 + 7.32i)T + (-40.1 - 8.52i)T^{2} \) |
| 43 | \( 1 + (-2.26 + 1.01i)T + (28.7 - 31.9i)T^{2} \) |
| 47 | \( 1 + (-2.33 + 1.69i)T + (14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (-6.65 + 1.41i)T + (48.4 - 21.5i)T^{2} \) |
| 59 | \( 1 + (0.391 + 3.72i)T + (-57.7 + 12.2i)T^{2} \) |
| 61 | \( 1 + 1.28T + 61T^{2} \) |
| 67 | \( 1 + (3.45 + 5.97i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-1.60 + 0.341i)T + (64.8 - 28.8i)T^{2} \) |
| 73 | \( 1 + (4.17 - 4.63i)T + (-7.63 - 72.6i)T^{2} \) |
| 79 | \( 1 + (-1.13 - 1.25i)T + (-8.25 + 78.5i)T^{2} \) |
| 83 | \( 1 + (0.538 - 5.12i)T + (-81.1 - 17.2i)T^{2} \) |
| 89 | \( 1 + (4.92 + 15.1i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (-2.72 - 8.39i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−9.648190661718776506259035806576, −8.811255008814327188510197960991, −8.404622800250397345525916126684, −7.34741579881452395662553858391, −6.42247022958222028960974555174, −5.70957909732601447606588227567, −4.06639158214931596710053126942, −3.36545410871121519470358897086, −1.80544407726287788063549432677, −0.70844047245994212259204756573,
1.37992437616228560558471898374, 3.13161013573521493513737203322, 4.00770137930752757304938670917, 5.27606420179140959653531327417, 6.11307376935686631669247813394, 7.03334595501335074037307676076, 7.77796083662667341375112108129, 8.880827206483667141330107321325, 9.399908404781619231045560033431, 10.24568373289633148344777033907