L(s) = 1 | + (−2.98 − 0.323i)3-s + 2.23i·5-s + 4.72·7-s + (8.79 + 1.92i)9-s + 4.76i·11-s + 1.06·13-s + (0.722 − 6.66i)15-s + 26.7i·17-s + 8.12·19-s + (−14.0 − 1.52i)21-s − 40.0i·23-s − 5.00·25-s + (−25.5 − 8.59i)27-s − 20.8i·29-s − 33.7·31-s + ⋯ |
L(s) = 1 | + (−0.994 − 0.107i)3-s + 0.447i·5-s + 0.675·7-s + (0.976 + 0.214i)9-s + 0.433i·11-s + 0.0820·13-s + (0.0481 − 0.444i)15-s + 1.57i·17-s + 0.427·19-s + (−0.671 − 0.0727i)21-s − 1.74i·23-s − 0.200·25-s + (−0.948 − 0.318i)27-s − 0.719i·29-s − 1.08·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.107 - 0.994i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.107 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.226544777\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.226544777\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (2.98 + 0.323i)T \) |
| 5 | \( 1 - 2.23iT \) |
good | 7 | \( 1 - 4.72T + 49T^{2} \) |
| 11 | \( 1 - 4.76iT - 121T^{2} \) |
| 13 | \( 1 - 1.06T + 169T^{2} \) |
| 17 | \( 1 - 26.7iT - 289T^{2} \) |
| 19 | \( 1 - 8.12T + 361T^{2} \) |
| 23 | \( 1 + 40.0iT - 529T^{2} \) |
| 29 | \( 1 + 20.8iT - 841T^{2} \) |
| 31 | \( 1 + 33.7T + 961T^{2} \) |
| 37 | \( 1 - 60.4T + 1.36e3T^{2} \) |
| 41 | \( 1 - 59.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 56.4T + 1.84e3T^{2} \) |
| 47 | \( 1 + 9.68iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 93.1iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 17.4iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 57.7T + 3.72e3T^{2} \) |
| 67 | \( 1 + 101.T + 4.48e3T^{2} \) |
| 71 | \( 1 - 90.1iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 40.0T + 5.32e3T^{2} \) |
| 79 | \( 1 - 65.3T + 6.24e3T^{2} \) |
| 83 | \( 1 - 117. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 119. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 15.2T + 9.40e3T^{2} \) |
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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
−10.29096540979080309176605063086, −9.347195814498791381858284403413, −8.120510523863727275694622254786, −7.50941803473770689976641972218, −6.40159841789269913136850775241, −5.91259112737058304259976722167, −4.71029587204923242360797396128, −4.04647884256891935492299756756, −2.38813310661247037326837642030, −1.17617652429699570808282846065,
0.51774046092520135337721673955, 1.67323506444249052988104771211, 3.37708429892297758973222276173, 4.57182684812668799795278967949, 5.29549947296686664784232945679, 5.92927506968693485268645872819, 7.24279991629875244720906436266, 7.68849263401784855464385842080, 9.074459395538952095548026536775, 9.530858350361376614160374118363