L(s) = 1 | + (−1.42 + 0.987i)3-s + (−0.679 − 0.392i)5-s + (−2.49 − 0.891i)7-s + (1.04 − 2.81i)9-s + (−1.22 + 0.708i)11-s + (−1.50 + 0.868i)13-s + (1.35 − 0.112i)15-s + (5.43 + 3.13i)17-s + (0.736 + 1.27i)19-s + (4.42 − 1.19i)21-s + (−4.85 − 2.80i)23-s + (−2.19 − 3.79i)25-s + (1.28 + 5.03i)27-s + (3.95 − 6.85i)29-s + 8.41·31-s + ⋯ |
L(s) = 1 | + (−0.821 + 0.570i)3-s + (−0.303 − 0.175i)5-s + (−0.941 − 0.337i)7-s + (0.349 − 0.936i)9-s + (−0.369 + 0.213i)11-s + (−0.417 + 0.240i)13-s + (0.349 − 0.0291i)15-s + (1.31 + 0.761i)17-s + (0.168 + 0.292i)19-s + (0.965 − 0.259i)21-s + (−1.01 − 0.584i)23-s + (−0.438 − 0.759i)25-s + (0.246 + 0.969i)27-s + (0.734 − 1.27i)29-s + 1.51·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.955 - 0.293i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.955 - 0.293i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8868395809\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8868395809\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.42 - 0.987i)T \) |
| 7 | \( 1 + (2.49 + 0.891i)T \) |
good | 5 | \( 1 + (0.679 + 0.392i)T + (2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (1.22 - 0.708i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (1.50 - 0.868i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-5.43 - 3.13i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.736 - 1.27i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (4.85 + 2.80i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.95 + 6.85i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 8.41T + 31T^{2} \) |
| 37 | \( 1 + (-3.74 - 6.48i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-7.19 + 4.15i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-7.85 - 4.53i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 0.110T + 47T^{2} \) |
| 53 | \( 1 + (4.28 - 7.42i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + 0.0736T + 59T^{2} \) |
| 61 | \( 1 + 1.23iT - 61T^{2} \) |
| 67 | \( 1 - 11.8iT - 67T^{2} \) |
| 71 | \( 1 + 0.390iT - 71T^{2} \) |
| 73 | \( 1 + (-3.70 - 2.13i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 - 6.00iT - 79T^{2} \) |
| 83 | \( 1 + (-7.88 + 13.6i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-6.15 + 3.55i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-5.89 - 3.40i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.03421432423860410247501645211, −9.592398436328969752746109645036, −8.256732183541885237222039066951, −7.51896365630739432483557493328, −6.24240265740127936153522344813, −5.96663991323598717685861347823, −4.55763248510749231480551194104, −4.02007942214656696375496435621, −2.77382936747975833338098947966, −0.75292591998861548449796846618,
0.74945409087265043503886640068, 2.48821227394352426141027910778, 3.50299803290762548727869389664, 4.94129336210880328131693941612, 5.70851224254883851863151140036, 6.46176625097283681399236946609, 7.46642497833345745920262758119, 7.898992887863212510684737059507, 9.279936966195200120940059333765, 9.985022821893899769779655508553