L(s) = 1 | + (−0.351 + 1.69i)3-s + (2.22 + 0.184i)5-s + (−2.84 + 2.84i)7-s + (−2.75 − 1.19i)9-s + (4.71 + 1.53i)11-s + (−0.188 + 0.369i)13-s + (−1.09 + 3.71i)15-s + (−7.50 − 1.18i)17-s + (2.68 + 3.69i)19-s + (−3.81 − 5.81i)21-s + (0.153 + 0.301i)23-s + (4.93 + 0.821i)25-s + (2.98 − 4.25i)27-s + (0.836 + 0.607i)29-s + (−2.24 + 1.63i)31-s + ⋯ |
L(s) = 1 | + (−0.202 + 0.979i)3-s + (0.996 + 0.0824i)5-s + (−1.07 + 1.07i)7-s + (−0.917 − 0.397i)9-s + (1.42 + 0.461i)11-s + (−0.0522 + 0.102i)13-s + (−0.282 + 0.959i)15-s + (−1.82 − 0.288i)17-s + (0.616 + 0.848i)19-s + (−0.833 − 1.26i)21-s + (0.0320 + 0.0628i)23-s + (0.986 + 0.164i)25-s + (0.575 − 0.818i)27-s + (0.155 + 0.112i)29-s + (−0.403 + 0.292i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.194 - 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.194 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.779850 + 0.950123i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.779850 + 0.950123i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.351 - 1.69i)T \) |
| 5 | \( 1 + (-2.22 - 0.184i)T \) |
good | 7 | \( 1 + (2.84 - 2.84i)T - 7iT^{2} \) |
| 11 | \( 1 + (-4.71 - 1.53i)T + (8.89 + 6.46i)T^{2} \) |
| 13 | \( 1 + (0.188 - 0.369i)T + (-7.64 - 10.5i)T^{2} \) |
| 17 | \( 1 + (7.50 + 1.18i)T + (16.1 + 5.25i)T^{2} \) |
| 19 | \( 1 + (-2.68 - 3.69i)T + (-5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (-0.153 - 0.301i)T + (-13.5 + 18.6i)T^{2} \) |
| 29 | \( 1 + (-0.836 - 0.607i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (2.24 - 1.63i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (-4.31 - 2.20i)T + (21.7 + 29.9i)T^{2} \) |
| 41 | \( 1 + (-5.67 + 1.84i)T + (33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + (-2.46 - 2.46i)T + 43iT^{2} \) |
| 47 | \( 1 + (1.21 + 7.64i)T + (-44.6 + 14.5i)T^{2} \) |
| 53 | \( 1 + (-8.21 + 1.30i)T + (50.4 - 16.3i)T^{2} \) |
| 59 | \( 1 + (3.09 + 9.53i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (-3.00 + 9.24i)T + (-49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 + (-1.15 + 7.29i)T + (-63.7 - 20.7i)T^{2} \) |
| 71 | \( 1 + (1.25 - 1.73i)T + (-21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-12.7 + 6.49i)T + (42.9 - 59.0i)T^{2} \) |
| 79 | \( 1 + (4.46 - 6.14i)T + (-24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (1.02 - 6.47i)T + (-78.9 - 25.6i)T^{2} \) |
| 89 | \( 1 + (-4.13 + 12.7i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (-2.19 + 0.348i)T + (92.2 - 29.9i)T^{2} \) |
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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
−11.94412612459506217541653159495, −11.01807855117824876588893954605, −9.807347304831909025045914841873, −9.369192531956295453034655530786, −8.776092021889220612398779262384, −6.66218616123791976780099518451, −6.12778114128889626509448439596, −5.01450078023512773290743282193, −3.67930430564347491587031767549, −2.32517025008377050531681469720,
0.976123706203028559119353625361, 2.61972046978436080712827878572, 4.19825589814840824301880164268, 5.89549412605860187882827483301, 6.59989095829010026350705082242, 7.21414342454189292617610044835, 8.830739229922597592329319463229, 9.439014941559842656955709379388, 10.67980736934001713235796440544, 11.43959655921057668980582700515