L(s) = 1 | + (2.60 + 0.763i)3-s + (−0.841 + 0.540i)5-s + (0.742 − 5.16i)7-s + (3.65 + 2.35i)9-s + (2.19 + 4.81i)11-s + (−0.380 − 2.64i)13-s + (−2.60 + 0.763i)15-s + (2.62 − 3.02i)17-s + (3.33 + 3.84i)19-s + (5.87 − 12.8i)21-s + (−4.55 + 1.49i)23-s + (0.415 − 0.909i)25-s + (2.39 + 2.76i)27-s + (−5.18 + 5.98i)29-s + (−0.294 + 0.0864i)31-s + ⋯ |
L(s) = 1 | + (1.50 + 0.440i)3-s + (−0.376 + 0.241i)5-s + (0.280 − 1.95i)7-s + (1.21 + 0.783i)9-s + (0.663 + 1.45i)11-s + (−0.105 − 0.733i)13-s + (−0.671 + 0.197i)15-s + (0.635 − 0.733i)17-s + (0.764 + 0.881i)19-s + (1.28 − 2.80i)21-s + (−0.949 + 0.312i)23-s + (0.0830 − 0.181i)25-s + (0.460 + 0.531i)27-s + (−0.962 + 1.11i)29-s + (−0.0528 + 0.0155i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0573i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0573i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.23638 + 0.0642346i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.23638 + 0.0642346i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.841 - 0.540i)T \) |
| 23 | \( 1 + (4.55 - 1.49i)T \) |
good | 3 | \( 1 + (-2.60 - 0.763i)T + (2.52 + 1.62i)T^{2} \) |
| 7 | \( 1 + (-0.742 + 5.16i)T + (-6.71 - 1.97i)T^{2} \) |
| 11 | \( 1 + (-2.19 - 4.81i)T + (-7.20 + 8.31i)T^{2} \) |
| 13 | \( 1 + (0.380 + 2.64i)T + (-12.4 + 3.66i)T^{2} \) |
| 17 | \( 1 + (-2.62 + 3.02i)T + (-2.41 - 16.8i)T^{2} \) |
| 19 | \( 1 + (-3.33 - 3.84i)T + (-2.70 + 18.8i)T^{2} \) |
| 29 | \( 1 + (5.18 - 5.98i)T + (-4.12 - 28.7i)T^{2} \) |
| 31 | \( 1 + (0.294 - 0.0864i)T + (26.0 - 16.7i)T^{2} \) |
| 37 | \( 1 + (0.239 + 0.154i)T + (15.3 + 33.6i)T^{2} \) |
| 41 | \( 1 + (2.42 - 1.56i)T + (17.0 - 37.2i)T^{2} \) |
| 43 | \( 1 + (1.15 + 0.339i)T + (36.1 + 23.2i)T^{2} \) |
| 47 | \( 1 + 8.10T + 47T^{2} \) |
| 53 | \( 1 + (0.852 - 5.92i)T + (-50.8 - 14.9i)T^{2} \) |
| 59 | \( 1 + (1.14 + 7.93i)T + (-56.6 + 16.6i)T^{2} \) |
| 61 | \( 1 + (-8.37 + 2.45i)T + (51.3 - 32.9i)T^{2} \) |
| 67 | \( 1 + (-1.58 + 3.47i)T + (-43.8 - 50.6i)T^{2} \) |
| 71 | \( 1 + (-0.519 + 1.13i)T + (-46.4 - 53.6i)T^{2} \) |
| 73 | \( 1 + (2.65 + 3.05i)T + (-10.3 + 72.2i)T^{2} \) |
| 79 | \( 1 + (-0.818 - 5.69i)T + (-75.7 + 22.2i)T^{2} \) |
| 83 | \( 1 + (1.06 + 0.681i)T + (34.4 + 75.4i)T^{2} \) |
| 89 | \( 1 + (-14.0 - 4.13i)T + (74.8 + 48.1i)T^{2} \) |
| 97 | \( 1 + (8.72 - 5.60i)T + (40.2 - 88.2i)T^{2} \) |
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
−10.76077115558820029057386114460, −9.861481985461325900707783042449, −9.618597627475434248324946749853, −8.060413050649067133210482798332, −7.60907036809425037459365469205, −6.95339861159347932366973503412, −4.90820479743024316726802631848, −3.88266032310549801636500608096, −3.36948176345755120469381666927, −1.61063097391017270900497216373,
1.79989970064410434841334938951, 2.86107023438925232468242933942, 3.84206624657165227094687622779, 5.44894775785988541177357538226, 6.40873693879739445484034927196, 7.81001019754311632586765328475, 8.483061333063755611886843741658, 8.939633750216665361389464555914, 9.669283946675154448094561174479, 11.49573918000825746527613319649