L(s) = 1 | + (−0.707 − 0.707i)2-s + (−1.49 + 0.867i)3-s + 1.00i·4-s + (−0.850 + 2.06i)5-s + (1.67 + 0.446i)6-s + (0.811 − 0.811i)7-s + (0.707 − 0.707i)8-s + (1.49 − 2.60i)9-s + (2.06 − 0.861i)10-s − 1.25i·11-s + (−0.867 − 1.49i)12-s + (3.40 + 3.40i)13-s − 1.14·14-s + (−0.519 − 3.83i)15-s − 1.00·16-s + (3.04 + 3.04i)17-s + ⋯ |
L(s) = 1 | + (−0.499 − 0.499i)2-s + (−0.865 + 0.500i)3-s + 0.500i·4-s + (−0.380 + 0.924i)5-s + (0.683 + 0.182i)6-s + (0.306 − 0.306i)7-s + (0.250 − 0.250i)8-s + (0.498 − 0.867i)9-s + (0.652 − 0.272i)10-s − 0.377i·11-s + (−0.250 − 0.432i)12-s + (0.944 + 0.944i)13-s − 0.306·14-s + (−0.134 − 0.990i)15-s − 0.250·16-s + (0.737 + 0.737i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.358 - 0.933i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.358 - 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.323858 + 0.471171i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.323858 + 0.471171i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 + 0.707i)T \) |
| 3 | \( 1 + (1.49 - 0.867i)T \) |
| 5 | \( 1 + (0.850 - 2.06i)T \) |
| 19 | \( 1 + iT \) |
good | 7 | \( 1 + (-0.811 + 0.811i)T - 7iT^{2} \) |
| 11 | \( 1 + 1.25iT - 11T^{2} \) |
| 13 | \( 1 + (-3.40 - 3.40i)T + 13iT^{2} \) |
| 17 | \( 1 + (-3.04 - 3.04i)T + 17iT^{2} \) |
| 23 | \( 1 + (5.51 - 5.51i)T - 23iT^{2} \) |
| 29 | \( 1 + 5.49T + 29T^{2} \) |
| 31 | \( 1 + 5.80T + 31T^{2} \) |
| 37 | \( 1 + (7.08 - 7.08i)T - 37iT^{2} \) |
| 41 | \( 1 - 0.259iT - 41T^{2} \) |
| 43 | \( 1 + (-5.60 - 5.60i)T + 43iT^{2} \) |
| 47 | \( 1 + (6.12 + 6.12i)T + 47iT^{2} \) |
| 53 | \( 1 + (-0.465 + 0.465i)T - 53iT^{2} \) |
| 59 | \( 1 - 9.08T + 59T^{2} \) |
| 61 | \( 1 - 6.32T + 61T^{2} \) |
| 67 | \( 1 + (2.40 - 2.40i)T - 67iT^{2} \) |
| 71 | \( 1 - 0.747iT - 71T^{2} \) |
| 73 | \( 1 + (-5.45 - 5.45i)T + 73iT^{2} \) |
| 79 | \( 1 + 4.84iT - 79T^{2} \) |
| 83 | \( 1 + (-4.22 + 4.22i)T - 83iT^{2} \) |
| 89 | \( 1 + 12.7T + 89T^{2} \) |
| 97 | \( 1 + (12.5 - 12.5i)T - 97iT^{2} \) |
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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
−11.19236167286420447111119608755, −10.27094546172298223007930521433, −9.579124226742172752174918445011, −8.440459652770288678617186801345, −7.43866931368367150817198021667, −6.51817605748168685729045785301, −5.57518916724531652107399585050, −4.01329999199069436314112243133, −3.52817909726844804737161727584, −1.58055556701626317439317472827,
0.45028974503364610714366331082, 1.82005787455640812159744978137, 4.02089756114522458615616182964, 5.36487405122997294591604411873, 5.65476677540505923625333047232, 6.97962987003324521945302972457, 7.85551387147610957135945642951, 8.447714996425253350671063316698, 9.518299072979725228131821517066, 10.52440981794383845051709239041