| L(s) = 1 | − 2-s + 4-s + (0.526 + 2.17i)5-s − 8-s + (−0.526 − 2.17i)10-s + 4.41i·11-s + 4.94·13-s + 16-s + 5.22i·17-s + 4.97i·19-s + (0.526 + 2.17i)20-s − 4.41i·22-s + 7.71·23-s + (−4.44 + 2.28i)25-s − 4.94·26-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + (0.235 + 0.971i)5-s − 0.353·8-s + (−0.166 − 0.687i)10-s + 1.33i·11-s + 1.37·13-s + 0.250·16-s + 1.26i·17-s + 1.14i·19-s + (0.117 + 0.485i)20-s − 0.942i·22-s + 1.60·23-s + (−0.889 + 0.457i)25-s − 0.969·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.641 - 0.767i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.641 - 0.767i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.470030905\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.470030905\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-0.526 - 2.17i)T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 - 4.41iT - 11T^{2} \) |
| 13 | \( 1 - 4.94T + 13T^{2} \) |
| 17 | \( 1 - 5.22iT - 17T^{2} \) |
| 19 | \( 1 - 4.97iT - 19T^{2} \) |
| 23 | \( 1 - 7.71T + 23T^{2} \) |
| 29 | \( 1 - 10.2iT - 29T^{2} \) |
| 31 | \( 1 - 6.43iT - 31T^{2} \) |
| 37 | \( 1 + 9.25iT - 37T^{2} \) |
| 41 | \( 1 + 6.79T + 41T^{2} \) |
| 43 | \( 1 + 2.08iT - 43T^{2} \) |
| 47 | \( 1 + 7.83iT - 47T^{2} \) |
| 53 | \( 1 + 5.67T + 53T^{2} \) |
| 59 | \( 1 - 12.1T + 59T^{2} \) |
| 61 | \( 1 + 6.76iT - 61T^{2} \) |
| 67 | \( 1 - 2.40iT - 67T^{2} \) |
| 71 | \( 1 + 10.4iT - 71T^{2} \) |
| 73 | \( 1 - 0.202T + 73T^{2} \) |
| 79 | \( 1 + 0.736T + 79T^{2} \) |
| 83 | \( 1 - 3.96iT - 83T^{2} \) |
| 89 | \( 1 - 6.61T + 89T^{2} \) |
| 97 | \( 1 - 14.5T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.663971745175531535379709097459, −7.939018940139214188573704563325, −6.94327270430449552326329671101, −6.82765204189997990499779431558, −5.86160770526853830104265015654, −5.07713769359982898998907027155, −3.72597921696628171436857789702, −3.32013709592483574130948986467, −1.98608765163177993924207538587, −1.44458066341253675568204052116,
0.60729974572789431677093804583, 1.11486575251402837394720533671, 2.51850451507748631502175407279, 3.29915580452947651692498296660, 4.41273465093176279302779285536, 5.21620005887353854924284488627, 6.01602880521626472134773952404, 6.58117312772558481549839522567, 7.57551649797875872552252339404, 8.369501990620008755488323245925
