L(s) = 1 | + (−1.62 − 0.608i)3-s + (1.53 + 0.886i)5-s + (2.64 − 0.0987i)7-s + (2.25 + 1.97i)9-s + (−5.33 + 3.08i)11-s + (−4.60 + 2.65i)13-s + (−1.95 − 2.37i)15-s + (4.69 + 2.71i)17-s + (−0.935 − 1.62i)19-s + (−4.34 − 1.44i)21-s + (0.562 + 0.324i)23-s + (−0.928 − 1.60i)25-s + (−2.46 − 4.57i)27-s + (−1.14 + 1.98i)29-s − 10.2·31-s + ⋯ |
L(s) = 1 | + (−0.936 − 0.351i)3-s + (0.686 + 0.396i)5-s + (0.999 − 0.0373i)7-s + (0.752 + 0.658i)9-s + (−1.60 + 0.929i)11-s + (−1.27 + 0.737i)13-s + (−0.503 − 0.612i)15-s + (1.13 + 0.657i)17-s + (−0.214 − 0.371i)19-s + (−0.948 − 0.316i)21-s + (0.117 + 0.0677i)23-s + (−0.185 − 0.321i)25-s + (−0.473 − 0.880i)27-s + (−0.213 + 0.369i)29-s − 1.83·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.313 - 0.949i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.313 - 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8527838300\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8527838300\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.62 + 0.608i)T \) |
| 7 | \( 1 + (-2.64 + 0.0987i)T \) |
good | 5 | \( 1 + (-1.53 - 0.886i)T + (2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (5.33 - 3.08i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (4.60 - 2.65i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-4.69 - 2.71i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.935 + 1.62i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.562 - 0.324i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (1.14 - 1.98i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 10.2T + 31T^{2} \) |
| 37 | \( 1 + (-1.09 - 1.89i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (6.64 - 3.83i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-1.07 - 0.620i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 0.468T + 47T^{2} \) |
| 53 | \( 1 + (0.941 - 1.63i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 - 9.24T + 59T^{2} \) |
| 61 | \( 1 - 6.33iT - 61T^{2} \) |
| 67 | \( 1 - 9.93iT - 67T^{2} \) |
| 71 | \( 1 - 11.1iT - 71T^{2} \) |
| 73 | \( 1 + (4.77 + 2.75i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 - 10.5iT - 79T^{2} \) |
| 83 | \( 1 + (4.06 - 7.03i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (0.228 - 0.132i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (12.5 + 7.26i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.15267359427920217218751382229, −9.856484218322635501176168296339, −8.358916983007458445368730496322, −7.39589445272769312007755359606, −7.07523025184474424691503795044, −5.68723829673650045231346310582, −5.21576561784420018058964270903, −4.37959448911004064944856706173, −2.46346961804501473914969598778, −1.70335164173498026031872851046,
0.41743909150968378715045712089, 1.99923949516808514644931638516, 3.38762827186232428012804179746, 4.95437417866844460501952014719, 5.31920979409725379669615562294, 5.81413228252294321575241076301, 7.39296880336702335198996135550, 7.85624294060979457306271861974, 9.030457637032209562473573807465, 9.952080300501676576852289874223