L(s) = 1 | − 4·2-s + (1.66 + 15.4i)3-s + 16·4-s − 65.2i·5-s + (−6.65 − 61.9i)6-s − 43.9·7-s − 64·8-s + (−237. + 51.5i)9-s + 261. i·10-s − 184.·11-s + (26.6 + 247. i)12-s + 483. i·13-s + 175.·14-s + (1.01e3 − 108. i)15-s + 256·16-s + 1.66e3i·17-s + ⋯ |
L(s) = 1 | − 0.707·2-s + (0.106 + 0.994i)3-s + 0.5·4-s − 1.16i·5-s + (−0.0754 − 0.703i)6-s − 0.339·7-s − 0.353·8-s + (−0.977 + 0.212i)9-s + 0.825i·10-s − 0.459·11-s + (0.0533 + 0.497i)12-s + 0.792i·13-s + 0.239·14-s + (1.16 − 0.124i)15-s + 0.250·16-s + 1.39i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0765 + 0.997i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.0765 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.4939275139\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4939275139\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 4T \) |
| 3 | \( 1 + (-1.66 - 15.4i)T \) |
| 59 | \( 1 + (2.62e4 + 4.88e3i)T \) |
good | 5 | \( 1 + 65.2iT - 3.12e3T^{2} \) |
| 7 | \( 1 + 43.9T + 1.68e4T^{2} \) |
| 11 | \( 1 + 184.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 483. iT - 3.71e5T^{2} \) |
| 17 | \( 1 - 1.66e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 464.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.98e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 6.84e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 - 9.47e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 - 4.96e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 + 1.88e4iT - 1.15e8T^{2} \) |
| 43 | \( 1 - 3.62e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 7.49e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 8.88e3iT - 4.18e8T^{2} \) |
| 61 | \( 1 + 3.76e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 - 2.94e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 4.55e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 + 7.20e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 3.62e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 2.57e3T + 3.93e9T^{2} \) |
| 89 | \( 1 + 7.28e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 3.00e4iT - 8.58e9T^{2} \) |
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
−10.24651233830198589392739980592, −9.358391388598759795270877582742, −8.745404359548380721896328426098, −8.015365945088739228003102268405, −6.53538956848268587326738568613, −5.36358869921610467858935493601, −4.44502278251158641422824020445, −3.21520771040132455696300898673, −1.65723072204058247214126457291, −0.17886893602167517158052805099,
1.03027316883371640795882280825, 2.66367439055619432490064341675, 3.06686031803362855964023411171, 5.36182822656451861022167995538, 6.48164397752508315203002087490, 7.22689097799207950054966990270, 7.80717032988482251956492366482, 9.005338089996605906172103781410, 9.942401181419689506329904318795, 10.99820341632534944296200634602