L(s) = 1 | − 2-s + (−0.554 + 1.64i)3-s + 4-s + (−0.5 + 0.866i)5-s + (0.554 − 1.64i)6-s + (2.32 − 1.26i)7-s − 8-s + (−2.38 − 1.82i)9-s + (0.5 − 0.866i)10-s + (−1.25 − 2.17i)11-s + (−0.554 + 1.64i)12-s + (−2.97 − 5.15i)13-s + (−2.32 + 1.26i)14-s + (−1.14 − 1.30i)15-s + 16-s + (−2.65 + 4.60i)17-s + ⋯ |
L(s) = 1 | − 0.707·2-s + (−0.320 + 0.947i)3-s + 0.5·4-s + (−0.223 + 0.387i)5-s + (0.226 − 0.669i)6-s + (0.878 − 0.478i)7-s − 0.353·8-s + (−0.794 − 0.606i)9-s + (0.158 − 0.273i)10-s + (−0.378 − 0.655i)11-s + (−0.160 + 0.473i)12-s + (−0.826 − 1.43i)13-s + (−0.621 + 0.338i)14-s + (−0.295 − 0.335i)15-s + 0.250·16-s + (−0.644 + 1.11i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.131 + 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.131 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.354982 - 0.310988i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.354982 - 0.310988i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + (0.554 - 1.64i)T \) |
| 5 | \( 1 + (0.5 - 0.866i)T \) |
| 7 | \( 1 + (-2.32 + 1.26i)T \) |
good | 11 | \( 1 + (1.25 + 2.17i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (2.97 + 5.15i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (2.65 - 4.60i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (2.97 + 5.15i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.678 - 1.17i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.10 + 5.37i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 6.16T + 31T^{2} \) |
| 37 | \( 1 + (-2.58 - 4.47i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (4.43 + 7.67i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.977 + 1.69i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 2.04T + 47T^{2} \) |
| 53 | \( 1 + (0.179 - 0.311i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + 2.86T + 59T^{2} \) |
| 61 | \( 1 - 3.82T + 61T^{2} \) |
| 67 | \( 1 - 12.9T + 67T^{2} \) |
| 71 | \( 1 - 3.52T + 71T^{2} \) |
| 73 | \( 1 + (-5.60 + 9.70i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + 14.7T + 79T^{2} \) |
| 83 | \( 1 + (2.54 - 4.39i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (0.367 + 0.636i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (6.19 - 10.7i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.61451270559045531086497646184, −9.718936468410748202562684327450, −8.533360095777450837298754540223, −8.065278615981632817266079945292, −6.94211461734179015662377094381, −5.80586757433346989844974360477, −4.86679431164296218681147637882, −3.72716586831903467338445632521, −2.49131979764861365838878050463, −0.32459048819994495453110198454,
1.63142191599702669803506430823, 2.40648433876455633147101083518, 4.51506148760880498109098087614, 5.37735521535239739603748940748, 6.65386449793851428194079089464, 7.32224291695607909719173499190, 8.157860640696354402713354179662, 8.888060091222764247543004367444, 9.795057244480259852692239375354, 11.02925781874498349783332504630