L(s) = 1 | − 2-s + 4-s + (−1.75 + 3.03i)5-s + (−2.63 + 0.222i)7-s − 8-s + (1.75 − 3.03i)10-s + (−3.20 + 5.54i)11-s + (−0.213 + 3.59i)13-s + (2.63 − 0.222i)14-s + 16-s − 4.67·17-s + (−2.61 − 4.53i)19-s + (−1.75 + 3.03i)20-s + (3.20 − 5.54i)22-s + 2.16·23-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.5·4-s + (−0.784 + 1.35i)5-s + (−0.996 + 0.0839i)7-s − 0.353·8-s + (0.554 − 0.960i)10-s + (−0.966 + 1.67i)11-s + (−0.0590 + 0.998i)13-s + (0.704 − 0.0593i)14-s + 0.250·16-s − 1.13·17-s + (−0.600 − 1.04i)19-s + (−0.392 + 0.679i)20-s + (0.683 − 1.18i)22-s + 0.452·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.166 + 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.166 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1415309502\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1415309502\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + (2.63 - 0.222i)T \) |
| 13 | \( 1 + (0.213 - 3.59i)T \) |
good | 5 | \( 1 + (1.75 - 3.03i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (3.20 - 5.54i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + 4.67T + 17T^{2} \) |
| 19 | \( 1 + (2.61 + 4.53i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 - 2.16T + 23T^{2} \) |
| 29 | \( 1 + (-1.23 - 2.13i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-4.46 - 7.72i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 3.89T + 37T^{2} \) |
| 41 | \( 1 + (5.09 + 8.82i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (1.19 - 2.06i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-2.44 + 4.22i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-1.05 - 1.82i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 - 11.7T + 59T^{2} \) |
| 61 | \( 1 + (4.67 + 8.10i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (3.64 - 6.31i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (2.79 - 4.83i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-4.23 - 7.32i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.893 + 1.54i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 2.59T + 83T^{2} \) |
| 89 | \( 1 + 7.00T + 89T^{2} \) |
| 97 | \( 1 + (4.92 - 8.53i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.07769705751952594322508158517, −9.187265107218567834834568785577, −8.444862657064199426666578425685, −7.18757141055919112752417853853, −7.00282374191909175167778329624, −6.50968912276725225235790782900, −4.93017485652248160432305807968, −3.97549098308964310728925860979, −2.79131447440192711376642739527, −2.23965420905423160985402510918,
0.093047059849126789892231493337, 0.822645018586192293473634654469, 2.63901269997120126674357619257, 3.56655260769415483596703925418, 4.60252564439029173135927829962, 5.73460929807365520654940618643, 6.26718606225801175954495200774, 7.58371455974270180833209423714, 8.298757238450543813147496072838, 8.535206991874300032832236038542