L(s) = 1 | + (−0.587 − 0.809i)2-s + (−0.309 + 0.951i)4-s + (−1.70 − 1.44i)5-s + 1.72i·7-s + (0.951 − 0.309i)8-s + (−0.166 + 2.22i)10-s + (−1.97 + 1.43i)11-s + (−1.74 + 2.40i)13-s + (1.39 − 1.01i)14-s + (−0.809 − 0.587i)16-s + (−3.18 + 1.03i)17-s + (−0.694 − 2.13i)19-s + (1.90 − 1.17i)20-s + (2.32 + 0.754i)22-s + (5.35 + 7.37i)23-s + ⋯ |
L(s) = 1 | + (−0.415 − 0.572i)2-s + (−0.154 + 0.475i)4-s + (−0.762 − 0.646i)5-s + 0.652i·7-s + (0.336 − 0.109i)8-s + (−0.0528 + 0.705i)10-s + (−0.595 + 0.432i)11-s + (−0.484 + 0.666i)13-s + (0.373 − 0.271i)14-s + (−0.202 − 0.146i)16-s + (−0.771 + 0.250i)17-s + (−0.159 − 0.490i)19-s + (0.425 − 0.262i)20-s + (0.495 + 0.160i)22-s + (1.11 + 1.53i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0238 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0238 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.322464 + 0.314872i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.322464 + 0.314872i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.587 + 0.809i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (1.70 + 1.44i)T \) |
good | 7 | \( 1 - 1.72iT - 7T^{2} \) |
| 11 | \( 1 + (1.97 - 1.43i)T + (3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (1.74 - 2.40i)T + (-4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (3.18 - 1.03i)T + (13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (0.694 + 2.13i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (-5.35 - 7.37i)T + (-7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (2.89 - 8.91i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (1.89 + 5.82i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (1.94 - 2.67i)T + (-11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (4.95 + 3.60i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 - 7.06iT - 43T^{2} \) |
| 47 | \( 1 + (4.74 + 1.54i)T + (38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (-6.51 - 2.11i)T + (42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (-0.147 - 0.107i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (4.16 - 3.02i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (11.3 - 3.67i)T + (54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (-3.51 + 10.8i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (3.95 + 5.44i)T + (-22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (-3.50 + 10.8i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (8.69 - 2.82i)T + (67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (-6.56 + 4.77i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (-9.40 - 3.05i)T + (78.4 + 57.0i)T^{2} \) |
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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
−11.43145383259650918140393620230, −10.52226598141174595520318677858, −9.159939176919641468350893826007, −9.001093102882277807445514219576, −7.75466442099991702351294029699, −7.00989664272243381973078910787, −5.33767691406201586313716034302, −4.49211498639220447791755919303, −3.19036534963291407526264812365, −1.77404131112848282278957877911,
0.31483236115905909042509069562, 2.68556387787147667823323273572, 4.03952480245351232186578415172, 5.17483057228628655282287134298, 6.48653498190210351055501452538, 7.23405979438107287092121120468, 8.024368786177645338138518271207, 8.821855263758046801606968882384, 10.22320430632109730418203042075, 10.61633871726821414508258211384