| L(s) = 1 | − 1.41i·2-s + 1.43i·3-s − 2.00·4-s + 2.03·6-s − 10.1·7-s + 2.82i·8-s + 6.93·9-s − 13.0i·11-s − 2.87i·12-s + 23.2i·13-s + 14.4i·14-s + 4.00·16-s + 28.2·17-s − 9.80i·18-s − 11.6i·19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 0.479i·3-s − 0.500·4-s + 0.339·6-s − 1.45·7-s + 0.353i·8-s + 0.770·9-s − 1.18i·11-s − 0.239i·12-s + 1.78i·13-s + 1.02i·14-s + 0.250·16-s + 1.66·17-s − 0.544i·18-s − 0.614i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.923 - 0.384i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.923 - 0.384i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.05097590237\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.05097590237\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 1.41iT \) |
| 5 | \( 1 \) |
| 23 | \( 1 + (15.0 + 17.3i)T \) |
| good | 3 | \( 1 - 1.43iT - 9T^{2} \) |
| 7 | \( 1 + 10.1T + 49T^{2} \) |
| 11 | \( 1 + 13.0iT - 121T^{2} \) |
| 13 | \( 1 - 23.2iT - 169T^{2} \) |
| 17 | \( 1 - 28.2T + 289T^{2} \) |
| 19 | \( 1 + 11.6iT - 361T^{2} \) |
| 29 | \( 1 + 42.4T + 841T^{2} \) |
| 31 | \( 1 - 18.7T + 961T^{2} \) |
| 37 | \( 1 - 1.14T + 1.36e3T^{2} \) |
| 41 | \( 1 + 72.8T + 1.68e3T^{2} \) |
| 43 | \( 1 + 4.96T + 1.84e3T^{2} \) |
| 47 | \( 1 - 0.813iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 26.7T + 2.80e3T^{2} \) |
| 59 | \( 1 + 94.0T + 3.48e3T^{2} \) |
| 61 | \( 1 - 74.5iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 80.0T + 4.48e3T^{2} \) |
| 71 | \( 1 + 83.5T + 5.04e3T^{2} \) |
| 73 | \( 1 + 8.98iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 80.2iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 94.6T + 6.88e3T^{2} \) |
| 89 | \( 1 + 136. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 2.32T + 9.40e3T^{2} \) |
| 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
−9.302016775987711388073601914675, −8.770494364049170648976574449869, −7.46022272480734952516417875383, −6.54481497078701567085388973883, −5.73368148751504813377029883047, −4.49146218690628195363782070059, −3.68224376924694025836634123241, −2.99513999427569789063935221642, −1.50851604066017563401492924747, −0.01592846131997205946292198456,
1.44737036745198918921706532455, 3.10164917139781639761387433589, 3.88040868294594658096963624344, 5.26984162198720754980221630348, 5.93622322252753353573226067520, 6.83412490594380116571146783845, 7.61257065408605832384855075693, 7.988241269046437995257522575446, 9.447246290536669689110653112949, 9.993186746274891518395422904274
