L(s) = 1 | + (−0.275 + 0.477i)3-s + i·5-s + (1.57 − 0.908i)7-s + (1.34 + 2.33i)9-s + (−0.759 − 0.438i)11-s + (−2.94 + 2.07i)13-s + (−0.477 − 0.275i)15-s + (2.47 + 4.28i)17-s + (−3.18 + 1.83i)19-s + 1.00i·21-s + (2.69 − 4.66i)23-s − 25-s − 3.14·27-s + (0.214 − 0.371i)29-s + 2.36i·31-s + ⋯ |
L(s) = 1 | + (−0.159 + 0.275i)3-s + 0.447i·5-s + (0.594 − 0.343i)7-s + (0.449 + 0.778i)9-s + (−0.229 − 0.132i)11-s + (−0.817 + 0.576i)13-s + (−0.123 − 0.0711i)15-s + (0.599 + 1.03i)17-s + (−0.729 + 0.421i)19-s + 0.218i·21-s + (0.562 − 0.973i)23-s − 0.200·25-s − 0.604·27-s + (0.0398 − 0.0690i)29-s + 0.424i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.103 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.103 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.397381183\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.397381183\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - iT \) |
| 13 | \( 1 + (2.94 - 2.07i)T \) |
good | 3 | \( 1 + (0.275 - 0.477i)T + (-1.5 - 2.59i)T^{2} \) |
| 7 | \( 1 + (-1.57 + 0.908i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (0.759 + 0.438i)T + (5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (-2.47 - 4.28i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (3.18 - 1.83i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.69 + 4.66i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.214 + 0.371i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 2.36iT - 31T^{2} \) |
| 37 | \( 1 + (-7.40 - 4.27i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.03 - 0.600i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (1.00 + 1.74i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 8.92iT - 47T^{2} \) |
| 53 | \( 1 + 5.45T + 53T^{2} \) |
| 59 | \( 1 + (3.96 - 2.29i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (0.952 + 1.64i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-4.42 - 2.55i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (13.4 - 7.76i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 - 10.7iT - 73T^{2} \) |
| 79 | \( 1 - 11.2T + 79T^{2} \) |
| 83 | \( 1 - 2.60iT - 83T^{2} \) |
| 89 | \( 1 + (-0.645 - 0.372i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-8.46 + 4.88i)T + (48.5 - 84.0i)T^{2} \) |
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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
−10.34544834993347655846895825315, −9.497075014641942648051271261608, −8.285478806939027066449381116099, −7.74893535722859888317105895278, −6.82182168197118648192814994347, −5.88024604099762592628398513914, −4.73912379024748217848318501022, −4.20561447742812891533708865635, −2.77251239090939786538350052790, −1.60546912566357794568833747138,
0.66089757776636581222029026397, 2.06886263005888142499309461867, 3.32953681441830716833729920623, 4.64310868832789374687087512437, 5.28412256457965263826324283193, 6.28244747792771087768971911743, 7.37905619069226572093472261210, 7.84698027299041008389871499378, 9.049282063468825955860159775546, 9.543165796983175664614550482133