| L(s) = 1 | + (−0.896 − 0.326i)2-s + (0.0971 + 0.266i)3-s + (−0.834 − 0.700i)4-s + (0.684 + 2.12i)5-s − 0.270i·6-s + (−0.0783 − 0.0138i)7-s + (1.47 + 2.55i)8-s + (2.23 − 1.87i)9-s + (0.0808 − 2.13i)10-s + (2.24 + 3.87i)11-s + (0.105 − 0.290i)12-s + (0.642 + 0.539i)13-s + (0.0657 + 0.0379i)14-s + (−0.501 + 0.389i)15-s + (−0.109 − 0.622i)16-s + (3.63 − 3.04i)17-s + ⋯ |
| L(s) = 1 | + (−0.633 − 0.230i)2-s + (0.0560 + 0.154i)3-s + (−0.417 − 0.350i)4-s + (0.306 + 0.951i)5-s − 0.110i·6-s + (−0.0296 − 0.00522i)7-s + (0.521 + 0.902i)8-s + (0.745 − 0.625i)9-s + (0.0255 − 0.674i)10-s + (0.675 + 1.16i)11-s + (0.0305 − 0.0839i)12-s + (0.178 + 0.149i)13-s + (0.0175 + 0.0101i)14-s + (−0.129 + 0.100i)15-s + (−0.0274 − 0.155i)16-s + (0.881 − 0.739i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.938 - 0.345i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.938 - 0.345i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.864310 + 0.153829i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.864310 + 0.153829i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-0.684 - 2.12i)T \) |
| 37 | \( 1 + (4.06 + 4.52i)T \) |
| good | 2 | \( 1 + (0.896 + 0.326i)T + (1.53 + 1.28i)T^{2} \) |
| 3 | \( 1 + (-0.0971 - 0.266i)T + (-2.29 + 1.92i)T^{2} \) |
| 7 | \( 1 + (0.0783 + 0.0138i)T + (6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (-2.24 - 3.87i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.642 - 0.539i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-3.63 + 3.04i)T + (2.95 - 16.7i)T^{2} \) |
| 19 | \( 1 + (-0.698 - 1.92i)T + (-14.5 + 12.2i)T^{2} \) |
| 23 | \( 1 + (-0.00227 + 0.00393i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (0.499 - 0.288i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 5.06iT - 31T^{2} \) |
| 41 | \( 1 + (4.55 + 3.81i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 - 1.01T + 43T^{2} \) |
| 47 | \( 1 + (6.88 + 3.97i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (0.788 - 0.138i)T + (49.8 - 18.1i)T^{2} \) |
| 59 | \( 1 + (-7.84 + 1.38i)T + (55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (2.86 - 3.40i)T + (-10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (-3.19 - 0.563i)T + (62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (-5.63 + 2.05i)T + (54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + 9.38iT - 73T^{2} \) |
| 79 | \( 1 + (6.85 + 1.20i)T + (74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (-9.01 - 10.7i)T + (-14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (-10.5 + 1.86i)T + (83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (-3.54 + 6.13i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−12.51998299734714310434861038394, −11.51759051377581421914041911597, −10.25331735052472768485045230266, −9.865767149538444811565317026539, −9.009101045710630438309227596513, −7.50420641388807355243411049244, −6.59670881758498919402213041740, −5.10401136674995304126821520790, −3.66113127761655466214619811810, −1.70737589497730084927000453693,
1.21787424267635319420405298217, 3.70541158132441799064668971090, 4.97315506079772433264831643777, 6.37165391763855877496941117265, 7.83271580880873353072080962851, 8.433028388088739567235960703134, 9.407608664177508304404182510135, 10.25092085971191042841170732747, 11.63208977826889937563779508040, 12.82262744222820374468345593709
