| L(s) = 1 | + (−1.5 − 0.866i)3-s + (3.54 − 2.04i)5-s + (−3.04 − 6.30i)7-s + (1.5 + 2.59i)9-s + (−10.0 + 17.3i)11-s − 21.9i·13-s − 7.08·15-s + (−10.9 − 6.29i)17-s + (−4.58 + 2.64i)19-s + (−0.894 + 12.0i)21-s + (8.75 + 15.1i)23-s + (−4.13 + 7.16i)25-s − 5.19i·27-s + 1.81·29-s + (−37.2 − 21.4i)31-s + ⋯ |
| L(s) = 1 | + (−0.5 − 0.288i)3-s + (0.708 − 0.409i)5-s + (−0.434 − 0.900i)7-s + (0.166 + 0.288i)9-s + (−0.912 + 1.58i)11-s − 1.68i·13-s − 0.472·15-s + (−0.641 − 0.370i)17-s + (−0.241 + 0.139i)19-s + (−0.0425 + 0.575i)21-s + (0.380 + 0.659i)23-s + (−0.165 + 0.286i)25-s − 0.192i·27-s + 0.0626·29-s + (−1.20 − 0.693i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.891 - 0.453i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.891 - 0.453i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1648159464\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1648159464\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.5 + 0.866i)T \) |
| 7 | \( 1 + (3.04 + 6.30i)T \) |
| good | 5 | \( 1 + (-3.54 + 2.04i)T + (12.5 - 21.6i)T^{2} \) |
| 11 | \( 1 + (10.0 - 17.3i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + 21.9iT - 169T^{2} \) |
| 17 | \( 1 + (10.9 + 6.29i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (4.58 - 2.64i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (-8.75 - 15.1i)T + (-264.5 + 458. i)T^{2} \) |
| 29 | \( 1 - 1.81T + 841T^{2} \) |
| 31 | \( 1 + (37.2 + 21.4i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (-19.1 - 33.2i)T + (-684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + 8.75iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 33.6T + 1.84e3T^{2} \) |
| 47 | \( 1 + (45.2 - 26.1i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (7.44 - 12.8i)T + (-1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (16.2 + 9.36i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (14.1 - 8.18i)T + (1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (56.5 - 98.0i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + 77.8T + 5.04e3T^{2} \) |
| 73 | \( 1 + (113. + 65.4i)T + (2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (56.9 + 98.6i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + 70.7iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (126. - 73.2i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + 22.0iT - 9.40e3T^{2} \) |
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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
−10.05462427188326987489593766130, −9.142789698909876165978874033093, −7.64524854468102297882523703484, −7.38061440866008713660486886204, −6.10877318483012376627000461492, −5.27778333963273283216085945706, −4.42804328739582793726820136525, −2.88356204959067200797023680350, −1.53910286087499362410519512019, −0.05973899386544489583686108973,
2.03522777088293019726830701054, 3.08683585290118084981077886983, 4.43965986236550904558884579575, 5.65684802397119631760591937014, 6.15252431372983239234759308185, 6.98193554922033425220636891406, 8.538008739560580899670444129661, 9.040107528675377893090502189068, 9.969644752741419803663439944088, 10.94859586559791066448418956627
