L(s) = 1 | + (−1.19 − 0.756i)2-s + (0.856 + 1.80i)4-s + 4.10i·5-s + 7-s + (0.343 − 2.80i)8-s + (3.10 − 4.90i)10-s − 2.67i·11-s + 3.02i·13-s + (−1.19 − 0.756i)14-s + (−2.53 + 3.09i)16-s + 5.12·17-s + 2.78i·19-s + (−7.41 + 3.51i)20-s + (−2.02 + 3.19i)22-s − 7.12·23-s + ⋯ |
L(s) = 1 | + (−0.845 − 0.534i)2-s + (0.428 + 0.903i)4-s + 1.83i·5-s + 0.377·7-s + (0.121 − 0.992i)8-s + (0.981 − 1.55i)10-s − 0.807i·11-s + 0.838i·13-s + (−0.319 − 0.202i)14-s + (−0.633 + 0.773i)16-s + 1.24·17-s + 0.637i·19-s + (−1.65 + 0.785i)20-s + (−0.431 + 0.682i)22-s − 1.48·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.121 - 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.121 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.652325 + 0.577334i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.652325 + 0.577334i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.19 + 0.756i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 5 | \( 1 - 4.10iT - 5T^{2} \) |
| 11 | \( 1 + 2.67iT - 11T^{2} \) |
| 13 | \( 1 - 3.02iT - 13T^{2} \) |
| 17 | \( 1 - 5.12T + 17T^{2} \) |
| 19 | \( 1 - 2.78iT - 19T^{2} \) |
| 23 | \( 1 + 7.12T + 23T^{2} \) |
| 29 | \( 1 - 8.83iT - 29T^{2} \) |
| 31 | \( 1 + 1.42T + 31T^{2} \) |
| 37 | \( 1 - 1.42iT - 37T^{2} \) |
| 41 | \( 1 + 5.12T + 41T^{2} \) |
| 43 | \( 1 - 2.39iT - 43T^{2} \) |
| 47 | \( 1 - 9.56T + 47T^{2} \) |
| 53 | \( 1 - 2.78iT - 53T^{2} \) |
| 59 | \( 1 + 4iT - 59T^{2} \) |
| 61 | \( 1 + 5.17iT - 61T^{2} \) |
| 67 | \( 1 + 0.244iT - 67T^{2} \) |
| 71 | \( 1 - 4.27T + 71T^{2} \) |
| 73 | \( 1 + 4.15T + 73T^{2} \) |
| 79 | \( 1 - 6.25T + 79T^{2} \) |
| 83 | \( 1 - 9.35iT - 83T^{2} \) |
| 89 | \( 1 + 11.2T + 89T^{2} \) |
| 97 | \( 1 - 6.69T + 97T^{2} \) |
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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
−10.95623409057398416159455216725, −10.35149481568703580864662485693, −9.618406022181595106139587239462, −8.390563780803850017617230491217, −7.59509921348481940258036280068, −6.79394240745581040817273494402, −5.86126396549973120084691332221, −3.82460985056023563629016957962, −3.08878082193135043807338288817, −1.82482043756576074841121446796,
0.69810166096243706579541683139, 1.98894732046040587315788481101, 4.27696268260208546521098781286, 5.26249218496061799532573904290, 5.90337893711496655524306614090, 7.51723939462532092463076526604, 8.044330232177693814165179140587, 8.832118761155878983713157536174, 9.718559286654454627348350139245, 10.28533231332801297061159328885