| 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 + (1.12 − 0.238i)7-s + (0.309 + 0.951i)8-s + (−0.978 − 0.207i)9-s + (0.104 + 0.994i)10-s + (−2.99 − 3.32i)11-s + (0.913 + 0.406i)12-s + (−0.205 + 0.0915i)13-s + (−0.767 + 0.851i)14-s + (0.809 + 0.587i)15-s + (−0.809 − 0.587i)16-s + (1.86 − 2.06i)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 + (0.423 − 0.0900i)7-s + (0.109 + 0.336i)8-s + (−0.326 − 0.0693i)9-s + (0.0330 + 0.314i)10-s + (−0.903 − 1.00i)11-s + (0.263 + 0.117i)12-s + (−0.0570 + 0.0254i)13-s + (−0.205 + 0.227i)14-s + (0.208 + 0.151i)15-s + (−0.202 − 0.146i)16-s + (0.451 − 0.501i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.698652 - 0.431832i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.698652 - 0.431832i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (3.07 - 4.64i)T \) |
| good | 7 | \( 1 + (-1.12 + 0.238i)T + (6.39 - 2.84i)T^{2} \) |
| 11 | \( 1 + (2.99 + 3.32i)T + (-1.14 + 10.9i)T^{2} \) |
| 13 | \( 1 + (0.205 - 0.0915i)T + (8.69 - 9.66i)T^{2} \) |
| 17 | \( 1 + (-1.86 + 2.06i)T + (-1.77 - 16.9i)T^{2} \) |
| 19 | \( 1 + (0.936 + 0.417i)T + (12.7 + 14.1i)T^{2} \) |
| 23 | \( 1 + (2.29 + 7.05i)T + (-18.6 + 13.5i)T^{2} \) |
| 29 | \( 1 + (2.29 - 1.66i)T + (8.96 - 27.5i)T^{2} \) |
| 37 | \( 1 + (4.17 + 7.23i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (0.598 + 5.69i)T + (-40.1 + 8.52i)T^{2} \) |
| 43 | \( 1 + (-4.33 - 1.93i)T + (28.7 + 31.9i)T^{2} \) |
| 47 | \( 1 + (8.06 + 5.86i)T + (14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (-6.10 - 1.29i)T + (48.4 + 21.5i)T^{2} \) |
| 59 | \( 1 + (-0.699 + 6.65i)T + (-57.7 - 12.2i)T^{2} \) |
| 61 | \( 1 - 6.58T + 61T^{2} \) |
| 67 | \( 1 + (-4.83 + 8.38i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-5.30 - 1.12i)T + (64.8 + 28.8i)T^{2} \) |
| 73 | \( 1 + (-5.63 - 6.26i)T + (-7.63 + 72.6i)T^{2} \) |
| 79 | \( 1 + (-5.87 + 6.52i)T + (-8.25 - 78.5i)T^{2} \) |
| 83 | \( 1 + (0.927 + 8.82i)T + (-81.1 + 17.2i)T^{2} \) |
| 89 | \( 1 + (0.556 - 1.71i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (0.298 - 0.917i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−9.912454354335208260416518476402, −8.927572706979342878804047719768, −8.419583502520507860721216591190, −7.58207843288479310319112156825, −6.47045387086761290764413411918, −5.45217684638845638884324535341, −4.93613940436548736206675797205, −3.58020270777079896175185704426, −2.21486501768234558000902186837, −0.46094623808724837476550820395,
1.57610186925334770196605308193, 2.41590266597781676075888778138, 3.66254746118423473002540134373, 5.02677995073750328363684234409, 5.99014940692389139011200791134, 7.10148815764659027587900723223, 7.75763864647862209680841984057, 8.371860880179221179543073184159, 9.672242235394829534955591778468, 10.03820839613288734518270793333
