L(s) = 1 | + (0.0726 − 0.0527i)2-s + (−0.309 + 0.951i)3-s + (−0.615 + 1.89i)4-s + (0.0277 + 0.0854i)6-s + 4.36·7-s + (0.110 + 0.341i)8-s + (−0.809 − 0.587i)9-s + (−3.55 + 2.58i)11-s + (−1.61 − 1.17i)12-s + (1.60 + 1.16i)13-s + (0.316 − 0.230i)14-s + (−3.19 − 2.32i)16-s + (0.308 + 0.948i)17-s − 0.0898·18-s + (0.417 + 1.28i)19-s + ⋯ |
L(s) = 1 | + (0.0513 − 0.0373i)2-s + (−0.178 + 0.549i)3-s + (−0.307 + 0.947i)4-s + (0.0113 + 0.0348i)6-s + 1.64·7-s + (0.0391 + 0.120i)8-s + (−0.269 − 0.195i)9-s + (−1.07 + 0.778i)11-s + (−0.465 − 0.337i)12-s + (0.444 + 0.323i)13-s + (0.0846 − 0.0615i)14-s + (−0.799 − 0.580i)16-s + (0.0747 + 0.229i)17-s − 0.0211·18-s + (0.0957 + 0.294i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.175 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.175 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.824950 + 0.985399i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.824950 + 0.985399i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.309 - 0.951i)T \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + (-0.0726 + 0.0527i)T + (0.618 - 1.90i)T^{2} \) |
| 7 | \( 1 - 4.36T + 7T^{2} \) |
| 11 | \( 1 + (3.55 - 2.58i)T + (3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (-1.60 - 1.16i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (-0.308 - 0.948i)T + (-13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (-0.417 - 1.28i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (1.90 - 1.38i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (2.46 - 7.58i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (1.13 + 3.49i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-1.16 - 0.844i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (-4.83 - 3.51i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 - 2.68T + 43T^{2} \) |
| 47 | \( 1 + (-3.38 + 10.4i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (-3.41 + 10.5i)T + (-42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (5.41 + 3.93i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-7.64 + 5.55i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (-3.99 - 12.2i)T + (-54.2 + 39.3i)T^{2} \) |
| 71 | \( 1 + (-2.26 + 6.96i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (-0.343 + 0.249i)T + (22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (1.96 - 6.04i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (-0.227 - 0.700i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (-7.91 + 5.75i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (-0.0104 + 0.0320i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−11.51229981519572558281364957470, −10.91487134078104525332162039980, −9.836180415309668623758228082491, −8.677075757847279903058430451036, −8.017380832665876833073059202560, −7.21081957935003572640959520036, −5.43882852891471522368544453897, −4.69483238893983754585536512176, −3.73029537537692144330059309782, −2.11281045513025671116268783912,
0.937509892255330249714474915915, 2.34930471001251065910968752903, 4.40039514692223320373617349121, 5.40181206180088925081757646444, 6.01011094829085546039094818703, 7.53815329286553843325541444641, 8.205600497612489917701102959884, 9.178567701448891942887944361588, 10.62058776980871532289914885041, 10.92407070709835699419712558363