L(s) = 1 | + (−1.32 − 1.11i)3-s + (−2.85 − 1.64i)5-s + 4.86·7-s + (0.494 + 2.95i)9-s + 4.65i·11-s + (0.131 − 0.0760i)13-s + (1.92 + 5.37i)15-s + (−2.37 − 1.36i)17-s + (1.11 + 4.21i)19-s + (−6.43 − 5.44i)21-s + (4.87 − 2.81i)23-s + (2.92 + 5.06i)25-s + (2.65 − 4.46i)27-s + (3.39 + 5.87i)29-s + 2.83i·31-s + ⋯ |
L(s) = 1 | + (−0.763 − 0.646i)3-s + (−1.27 − 0.736i)5-s + 1.84·7-s + (0.164 + 0.986i)9-s + 1.40i·11-s + (0.0365 − 0.0210i)13-s + (0.497 + 1.38i)15-s + (−0.575 − 0.332i)17-s + (0.256 + 0.966i)19-s + (−1.40 − 1.18i)21-s + (1.01 − 0.587i)23-s + (0.585 + 1.01i)25-s + (0.511 − 0.859i)27-s + (0.629 + 1.09i)29-s + 0.508i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.941 + 0.338i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.941 + 0.338i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.13356 - 0.197630i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.13356 - 0.197630i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.32 + 1.11i)T \) |
| 19 | \( 1 + (-1.11 - 4.21i)T \) |
good | 5 | \( 1 + (2.85 + 1.64i)T + (2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 - 4.86T + 7T^{2} \) |
| 11 | \( 1 - 4.65iT - 11T^{2} \) |
| 13 | \( 1 + (-0.131 + 0.0760i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (2.37 + 1.36i)T + (8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (-4.87 + 2.81i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.39 - 5.87i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 2.83iT - 31T^{2} \) |
| 37 | \( 1 + 5.14iT - 37T^{2} \) |
| 41 | \( 1 + (-0.166 + 0.287i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-2.12 + 3.68i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-7.83 + 4.52i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (0.486 + 0.842i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.30 + 5.72i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.39 - 2.42i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.73 + 2.15i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (7.75 - 13.4i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-3.13 + 5.42i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-8.38 - 4.84i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 4.12iT - 83T^{2} \) |
| 89 | \( 1 + (1.77 + 3.07i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (9.51 + 5.49i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.39105506472245549216582382440, −8.859063182882393754116937203166, −8.235273273551588513013678017654, −7.44842285228021335588966247073, −6.99934768143931622485118317787, −5.33932146313766419495743322481, −4.81501206466804747534800412847, −4.14916084472193311912604482215, −2.07937752547944517766490985653, −1.01700052709300284122424354034,
0.849728418084021186941086668153, 2.90848807286318108681813169868, 4.04083451952809822784801171813, 4.69861761715383048727719627269, 5.66320786048281104144562843221, 6.73006297389911665801292826456, 7.70035902603331874729118590027, 8.361149497459389695767045515201, 9.217486246141640740875584797103, 10.72870103594359869650776901289