| L(s) = 1 | − i·2-s − i·3-s − 4-s − 6-s − 4i·7-s + i·8-s − 9-s + i·12-s − 2i·13-s − 4·14-s + 16-s + 6i·17-s + i·18-s + 4·19-s − 4·21-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.577i·3-s − 0.5·4-s − 0.408·6-s − 1.51i·7-s + 0.353i·8-s − 0.333·9-s + 0.288i·12-s − 0.554i·13-s − 1.06·14-s + 0.250·16-s + 1.45i·17-s + 0.235i·18-s + 0.917·19-s − 0.872·21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.544708 - 0.881357i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.544708 - 0.881357i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 4iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 - 6iT - 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 2iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 + 18T + 89T^{2} \) |
| 97 | \( 1 - 2iT - 97T^{2} \) |
| 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
−12.73048483344986556804976084426, −11.72862706244732673515655909624, −10.59711490453538236750892836991, −10.02114576910288854614874100657, −8.428246467858918435959153272298, −7.52893777749673750646095101981, −6.24917907500364745105060801431, −4.56480584669636693327632140720, −3.23242613233288142627718190811, −1.19301600866377439702955307792,
2.87232562339919378192796826883, 4.72670202858721634233331620829, 5.61952237968355492352832454031, 6.85200004047624573112690645068, 8.303150793602142243576863666858, 9.156664271354851304940002344415, 9.941819242818724229118457896505, 11.57232332102479816450987379595, 12.16301303983113287659365630302, 13.65903418781624892598801003079
