| L(s) = 1 | − 1.56i·2-s − i·3-s − 0.438·4-s − 1.56·6-s + 5.12i·7-s − 2.43i·8-s − 9-s − 1.43·11-s + 0.438i·12-s + 2i·13-s + 8·14-s − 4.68·16-s − 7.12i·17-s + 1.56i·18-s − 5.12·19-s + ⋯ |
| L(s) = 1 | − 1.10i·2-s − 0.577i·3-s − 0.219·4-s − 0.637·6-s + 1.93i·7-s − 0.862i·8-s − 0.333·9-s − 0.433·11-s + 0.126i·12-s + 0.554i·13-s + 2.13·14-s − 1.17·16-s − 1.72i·17-s + 0.368i·18-s − 1.17·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9473941863\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9473941863\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 + 1.56iT - 2T^{2} \) |
| 7 | \( 1 - 5.12iT - 7T^{2} \) |
| 11 | \( 1 + 1.43T + 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 + 7.12iT - 17T^{2} \) |
| 19 | \( 1 + 5.12T + 19T^{2} \) |
| 23 | \( 1 + 6.56iT - 23T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 + 1.68iT - 37T^{2} \) |
| 41 | \( 1 + 1.68T + 41T^{2} \) |
| 43 | \( 1 + 7.68iT - 43T^{2} \) |
| 47 | \( 1 + 13.1iT - 47T^{2} \) |
| 53 | \( 1 - 3.43iT - 53T^{2} \) |
| 59 | \( 1 + 12T + 59T^{2} \) |
| 61 | \( 1 - 0.876T + 61T^{2} \) |
| 67 | \( 1 + 11.3iT - 67T^{2} \) |
| 71 | \( 1 + 2.87T + 71T^{2} \) |
| 73 | \( 1 + 1.68iT - 73T^{2} \) |
| 79 | \( 1 - 12T + 79T^{2} \) |
| 83 | \( 1 - 2.56iT - 83T^{2} \) |
| 89 | \( 1 + 12.2T + 89T^{2} \) |
| 97 | \( 1 + 5.68iT - 97T^{2} \) |
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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
−8.897093357618092536862461746904, −8.055700443637006929034412335553, −6.88718665361398093858988428751, −6.37172819632449471808324337488, −5.37289686107096121756035362722, −4.53099212439768909408156804900, −3.13864945625753608646767699428, −2.39994278312980442165263612156, −2.00275038207825084095357549378, −0.30466270150442596889614971048,
1.51057248231330080323362777799, 3.10094996730921783943123629256, 4.10453349609749920308058363473, 4.67115838437079923079693267772, 5.78147631065495380273705619340, 6.38886708310185818096181551809, 7.21114423886420853762153408035, 8.017176098629604895214388498313, 8.237231645014302324490897869735, 9.523890087761886198442740425968
