L(s) = 1 | + (−0.298 + 0.172i)3-s + (1.44 + 1.71i)5-s + (−1.75 − 1.01i)7-s + (−1.44 + 2.49i)9-s + (1.94 + 3.36i)11-s + (2.96 − 2.05i)13-s + (−0.725 − 0.262i)15-s + (−4.71 − 2.72i)17-s + (−2.94 + 5.09i)19-s + 0.700·21-s + (0.298 − 0.172i)23-s + (−0.850 + 4.92i)25-s − 2.02i·27-s + (1.5 + 2.59i)29-s − 1.18·31-s + ⋯ |
L(s) = 1 | + (−0.172 + 0.0996i)3-s + (0.644 + 0.764i)5-s + (−0.664 − 0.383i)7-s + (−0.480 + 0.831i)9-s + (0.585 + 1.01i)11-s + (0.821 − 0.570i)13-s + (−0.187 − 0.0678i)15-s + (−1.14 − 0.660i)17-s + (−0.674 + 1.16i)19-s + 0.152·21-s + (0.0623 − 0.0359i)23-s + (−0.170 + 0.985i)25-s − 0.390i·27-s + (0.278 + 0.482i)29-s − 0.212·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.503 - 0.864i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.503 - 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.144842811\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.144842811\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-1.44 - 1.71i)T \) |
| 13 | \( 1 + (-2.96 + 2.05i)T \) |
good | 3 | \( 1 + (0.298 - 0.172i)T + (1.5 - 2.59i)T^{2} \) |
| 7 | \( 1 + (1.75 + 1.01i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.94 - 3.36i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (4.71 + 2.72i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2.94 - 5.09i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.298 + 0.172i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.5 - 2.59i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 1.18T + 31T^{2} \) |
| 37 | \( 1 + (4.71 - 2.72i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (0.0902 + 0.156i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (1.15 + 0.669i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 12.2iT - 47T^{2} \) |
| 53 | \( 1 - 2.42iT - 53T^{2} \) |
| 59 | \( 1 + (-3.53 + 6.11i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (3.38 - 5.85i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (3.81 - 2.20i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (0.940 - 1.62i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 - 8.86iT - 73T^{2} \) |
| 79 | \( 1 - 11.1T + 79T^{2} \) |
| 83 | \( 1 + 7.83iT - 83T^{2} \) |
| 89 | \( 1 + (-6.12 - 10.6i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (5.02 + 2.90i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.33129688705208831364487031111, −9.525935907271224928925805724353, −8.632932286485085992384017142438, −7.57504375263217321354923868919, −6.68945558147305500898674090120, −6.13166876050631870330971985434, −5.07208347660390058591220423646, −3.96257931117751282025860915979, −2.86767596741634844516185830015, −1.77303967059882865097425424468,
0.50867349451479540649118703690, 1.99107244110755776837276699043, 3.34030067339823236069910267667, 4.31931195481641245490729975881, 5.54647591053456741538431634560, 6.38776226293503660473724812184, 6.60278790135604881634132899806, 8.430236692692469715411712910801, 8.955817425051042363261088557495, 9.251538393593171800216376390207