L(s) = 1 | + 3.29·3-s − i·5-s + (−2.01 + 1.71i)7-s + 7.85·9-s + 1.43i·11-s + 5.60i·13-s − 3.29i·15-s + 1.85i·17-s − 3.04·19-s + (−6.64 + 5.65i)21-s − 0.989i·23-s − 25-s + 16.0·27-s + 5.89·29-s + 7.82·31-s + ⋯ |
L(s) = 1 | + 1.90·3-s − 0.447i·5-s + (−0.761 + 0.648i)7-s + 2.61·9-s + 0.432i·11-s + 1.55i·13-s − 0.850i·15-s + 0.450i·17-s − 0.697·19-s + (−1.44 + 1.23i)21-s − 0.206i·23-s − 0.200·25-s + 3.08·27-s + 1.09·29-s + 1.40·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.761 - 0.648i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.761 - 0.648i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.366954492\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.366954492\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + iT \) |
| 7 | \( 1 + (2.01 - 1.71i)T \) |
good | 3 | \( 1 - 3.29T + 3T^{2} \) |
| 11 | \( 1 - 1.43iT - 11T^{2} \) |
| 13 | \( 1 - 5.60iT - 13T^{2} \) |
| 17 | \( 1 - 1.85iT - 17T^{2} \) |
| 19 | \( 1 + 3.04T + 19T^{2} \) |
| 23 | \( 1 + 0.989iT - 23T^{2} \) |
| 29 | \( 1 - 5.89T + 29T^{2} \) |
| 31 | \( 1 - 7.82T + 31T^{2} \) |
| 37 | \( 1 - 3.93T + 37T^{2} \) |
| 41 | \( 1 - 8.37iT - 41T^{2} \) |
| 43 | \( 1 - 0.890iT - 43T^{2} \) |
| 47 | \( 1 - 0.347T + 47T^{2} \) |
| 53 | \( 1 + 12.1T + 53T^{2} \) |
| 59 | \( 1 - 4.32T + 59T^{2} \) |
| 61 | \( 1 + 13.2iT - 61T^{2} \) |
| 67 | \( 1 + 15.3iT - 67T^{2} \) |
| 71 | \( 1 + 0.574iT - 71T^{2} \) |
| 73 | \( 1 - 13.6iT - 73T^{2} \) |
| 79 | \( 1 + 3.17iT - 79T^{2} \) |
| 83 | \( 1 + 7.59T + 83T^{2} \) |
| 89 | \( 1 - 8.08iT - 89T^{2} \) |
| 97 | \( 1 + 4.66iT - 97T^{2} \) |
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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
−9.107936123606531112979115012799, −8.378263328850732258695191877085, −7.967880024487965223952187387542, −6.72577122843171287211499205573, −6.41401868232627919357793997438, −4.68570514398953192510245798599, −4.22505458571405684978759224838, −3.17385549796772652908787692448, −2.39343793582761918065164248261, −1.57389112190683697171588991665,
0.954667450777090617190653112092, 2.54773766449097333669383398078, 2.99981153221213990465384269243, 3.73415146967024992819473965198, 4.59804166093889779166644606759, 6.00977614177082383667586436902, 6.90499128364019151679483380841, 7.55973259112728776739863898843, 8.221810673196098827362096234241, 8.802678278195641759560329615697