L(s) = 1 | − 1.73·3-s + (1.92 − 1.92i)5-s + (−3.58 − 3.58i)7-s + 2.99·9-s + (−6.83 − 6.83i)11-s + (4.86 − 12.0i)13-s + (−3.33 + 3.33i)15-s − 22.4i·17-s + (1.26 − 1.26i)19-s + (6.20 + 6.20i)21-s − 35.8i·23-s + 17.5i·25-s − 5.19·27-s + 5.29·29-s + (−22.8 + 22.8i)31-s + ⋯ |
L(s) = 1 | − 0.577·3-s + (0.385 − 0.385i)5-s + (−0.512 − 0.512i)7-s + 0.333·9-s + (−0.621 − 0.621i)11-s + (0.374 − 0.927i)13-s + (−0.222 + 0.222i)15-s − 1.32i·17-s + (0.0663 − 0.0663i)19-s + (0.295 + 0.295i)21-s − 1.55i·23-s + 0.703i·25-s − 0.192·27-s + 0.182·29-s + (−0.738 + 0.738i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0894 + 0.995i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0894 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.671786 - 0.734805i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.671786 - 0.734805i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + 1.73T \) |
| 13 | \( 1 + (-4.86 + 12.0i)T \) |
good | 5 | \( 1 + (-1.92 + 1.92i)T - 25iT^{2} \) |
| 7 | \( 1 + (3.58 + 3.58i)T + 49iT^{2} \) |
| 11 | \( 1 + (6.83 + 6.83i)T + 121iT^{2} \) |
| 17 | \( 1 + 22.4iT - 289T^{2} \) |
| 19 | \( 1 + (-1.26 + 1.26i)T - 361iT^{2} \) |
| 23 | \( 1 + 35.8iT - 529T^{2} \) |
| 29 | \( 1 - 5.29T + 841T^{2} \) |
| 31 | \( 1 + (22.8 - 22.8i)T - 961iT^{2} \) |
| 37 | \( 1 + (-27.7 - 27.7i)T + 1.36e3iT^{2} \) |
| 41 | \( 1 + (2.72 - 2.72i)T - 1.68e3iT^{2} \) |
| 43 | \( 1 - 53.4iT - 1.84e3T^{2} \) |
| 47 | \( 1 + (-24.5 - 24.5i)T + 2.20e3iT^{2} \) |
| 53 | \( 1 - 20.8T + 2.80e3T^{2} \) |
| 59 | \( 1 + (38.5 + 38.5i)T + 3.48e3iT^{2} \) |
| 61 | \( 1 - 19.7T + 3.72e3T^{2} \) |
| 67 | \( 1 + (6.69 - 6.69i)T - 4.48e3iT^{2} \) |
| 71 | \( 1 + (-35.7 + 35.7i)T - 5.04e3iT^{2} \) |
| 73 | \( 1 + (-68.0 - 68.0i)T + 5.32e3iT^{2} \) |
| 79 | \( 1 - 3.68T + 6.24e3T^{2} \) |
| 83 | \( 1 + (40.8 - 40.8i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 + (70.1 + 70.1i)T + 7.92e3iT^{2} \) |
| 97 | \( 1 + (-115. + 115. i)T - 9.40e3iT^{2} \) |
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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
−12.65241843537883890222075814353, −11.31115447513076232755308381313, −10.48478988307388109744342461666, −9.528909671927670943892316877546, −8.268472039326140032335863338981, −7.00033794372138662100565235845, −5.83334997035610206887307519275, −4.80557303906700220403322897866, −3.05130794560738177349655490434, −0.67747638676501171582990430968,
2.08012702184040378854789811703, 3.93811725606744623950354369659, 5.54503062709514896125542361305, 6.38365671892080083310916245021, 7.57161461480262758111555798906, 9.052820215836070416878825533535, 10.01657304104119437333121839527, 10.92405501356852604103174718483, 12.00850800397021322925753175595, 12.87318909067843669607329916693