L(s) = 1 | + (1 − 1.73i)5-s + (2 + 1.73i)7-s + (−2 − 3.46i)11-s − 13-s + (2 − 3.46i)19-s + (3 − 5.19i)23-s + (0.500 + 0.866i)25-s + (−2.5 − 4.33i)31-s + (5 − 1.73i)35-s + (−0.5 + 0.866i)37-s + 4·41-s − 43-s + (2 − 3.46i)47-s + (1.00 + 6.92i)49-s + (−3 − 5.19i)53-s + ⋯ |
L(s) = 1 | + (0.447 − 0.774i)5-s + (0.755 + 0.654i)7-s + (−0.603 − 1.04i)11-s − 0.277·13-s + (0.458 − 0.794i)19-s + (0.625 − 1.08i)23-s + (0.100 + 0.173i)25-s + (−0.449 − 0.777i)31-s + (0.845 − 0.292i)35-s + (−0.0821 + 0.142i)37-s + 0.624·41-s − 0.152·43-s + (0.291 − 0.505i)47-s + (0.142 + 0.989i)49-s + (−0.412 − 0.713i)53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.823501840\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.823501840\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-2 - 1.73i)T \) |
good | 5 | \( 1 + (-1 + 1.73i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (2 + 3.46i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + T + 13T^{2} \) |
| 17 | \( 1 + (-8.5 + 14.7i)T^{2} \) |
| 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 | \( 1 + (2.5 + 4.33i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (0.5 - 0.866i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 4T + 41T^{2} \) |
| 43 | \( 1 + T + 43T^{2} \) |
| 47 | \( 1 + (-2 + 3.46i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (3 + 5.19i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.5 + 2.59i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (5.5 + 9.52i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 14T + 71T^{2} \) |
| 73 | \( 1 + (-7 - 12.1i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (6.5 - 11.2i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 14T + 83T^{2} \) |
| 89 | \( 1 + (-3 + 5.19i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 9T + 97T^{2} \) |
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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
−9.142562256287727828682841364170, −8.602451448696083889421067042262, −7.947966547760633412314276520459, −6.90126094104091601505541024691, −5.77267080365393015011603722200, −5.24800358713264851021598547840, −4.51214234160735207134550349650, −3.07720860368886879697045209395, −2.11591438405784893882059971325, −0.75772461077159913566858037421,
1.44461821182599522797268036039, 2.48534898001024316624520542329, 3.59649481363211597308414624373, 4.69136893603325239482987738266, 5.41376898939079185091165290225, 6.47515135539313978584301072896, 7.48310748097923436102757240735, 7.60788226061128595362499722709, 8.892714363348812050889419324501, 9.823348706204214982254806881826