L(s) = 1 | + (1 − 1.73i)5-s + (−1 − 1.73i)11-s − 2·13-s + (3 + 5.19i)17-s + (−2 + 3.46i)19-s + (−3 + 5.19i)23-s + (0.500 + 0.866i)25-s + (−2 − 3.46i)31-s + (−5 + 8.66i)37-s − 2·41-s − 4·43-s + (2 − 3.46i)47-s + (6 + 10.3i)53-s − 3.99·55-s + (6 + 10.3i)59-s + ⋯ |
L(s) = 1 | + (0.447 − 0.774i)5-s + (−0.301 − 0.522i)11-s − 0.554·13-s + (0.727 + 1.26i)17-s + (−0.458 + 0.794i)19-s + (−0.625 + 1.08i)23-s + (0.100 + 0.173i)25-s + (−0.359 − 0.622i)31-s + (−0.821 + 1.42i)37-s − 0.312·41-s − 0.609·43-s + (0.291 − 0.505i)47-s + (0.824 + 1.42i)53-s − 0.539·55-s + (0.781 + 1.35i)59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.266 - 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.266 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.323230736\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.323230736\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (-1 + 1.73i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (1 + 1.73i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 + (-3 - 5.19i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2 - 3.46i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (3 - 5.19i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + (2 + 3.46i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (5 - 8.66i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + (-2 + 3.46i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-6 - 10.3i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-6 - 10.3i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-3 + 5.19i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-2 - 3.46i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 14T + 71T^{2} \) |
| 73 | \( 1 + (1 + 1.73i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-4 + 6.92i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 16T + 83T^{2} \) |
| 89 | \( 1 + (3 - 5.19i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 18T + 97T^{2} \) |
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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
−8.681527256123848456916835176163, −8.061994463765503089967364434994, −7.40466563012202686079955700986, −6.28760488521362731229621087702, −5.66938616783286731043020482749, −5.10688092101081822765516915254, −4.04876327244016960146224424850, −3.31285065771985867458379327118, −2.03615360149645158976430055265, −1.22384112172813856586386458748,
0.39515998465855833571210663598, 2.09736509449675253963484815553, 2.63980110593827424901677498510, 3.62441852098118416003424704394, 4.75932387608012061385097593670, 5.29026900721562383130903797779, 6.32461466273429206218829640833, 6.99294442133662316984318548358, 7.45652547754273956512006153005, 8.456915672812888172534004603436