L(s) = 1 | + (−0.309 + 0.535i)2-s + (−1.11 + 1.93i)3-s + (0.809 + 1.40i)4-s − 5-s + (−0.690 − 1.19i)6-s + (2.11 + 3.66i)7-s − 2.23·8-s + (−1 − 1.73i)9-s + (0.309 − 0.535i)10-s + (−0.118 + 0.204i)11-s − 3.61·12-s − 2.61·14-s + (1.11 − 1.93i)15-s + (−0.927 + 1.60i)16-s + (1.73 + 3.00i)17-s + 1.23·18-s + ⋯ |
L(s) = 1 | + (−0.218 + 0.378i)2-s + (−0.645 + 1.11i)3-s + (0.404 + 0.700i)4-s − 0.447·5-s + (−0.282 − 0.488i)6-s + (0.800 + 1.38i)7-s − 0.790·8-s + (−0.333 − 0.577i)9-s + (0.0977 − 0.169i)10-s + (−0.0355 + 0.0616i)11-s − 1.04·12-s − 0.699·14-s + (0.288 − 0.500i)15-s + (−0.231 + 0.401i)16-s + (0.421 + 0.729i)17-s + 0.291·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.859 + 0.511i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.859 + 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.287820 - 1.04725i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.287820 - 1.04725i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (0.309 - 0.535i)T + (-1 - 1.73i)T^{2} \) |
| 3 | \( 1 + (1.11 - 1.93i)T + (-1.5 - 2.59i)T^{2} \) |
| 7 | \( 1 + (-2.11 - 3.66i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (0.118 - 0.204i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (-1.73 - 3.00i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.11 - 3.66i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.88 - 3.25i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.73 + 6.47i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + (-1.5 + 2.59i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-5.97 + 10.3i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-3.11 - 5.40i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 4.94T + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 + (0.354 + 0.613i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (7.20 + 12.4i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (1.35 - 2.34i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (3.11 + 5.40i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 8.94T + 83T^{2} \) |
| 89 | \( 1 + (4.5 - 7.79i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (1.73 + 3.00i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.81308303907835108765258175071, −9.776733173857354274781714015356, −8.952768489495651845545116211183, −8.107013760266529085343730976508, −7.54993299220743744092741485971, −6.00784879807341044619043956536, −5.61186309718174485198965012310, −4.43567419913265968778768712635, −3.55163090941321183731862603101, −2.21081435980133761275670491911,
0.68669332897087479162575002553, 1.31506866949456603414308599452, 2.79672712140483236028118505322, 4.36633603800195727529310355161, 5.34476106833979977375902615982, 6.43197583303019262568765205480, 7.18580496167336078645866887550, 7.62189278834995121858423062145, 8.837903761111679287074311155867, 10.01933799698946758803849633744