| L(s) = 1 | + 2.23i·5-s + 7.68i·7-s − 8.06i·11-s + 14.0·13-s − 21.5i·17-s + 5.94i·19-s + (−16.1 + 16.3i)23-s − 5.00·25-s + 9.04·29-s − 34.1·31-s − 17.1·35-s + 55.1i·37-s + 42.2·41-s + 37.9i·43-s − 70.5·47-s + ⋯ |
| L(s) = 1 | + 0.447i·5-s + 1.09i·7-s − 0.733i·11-s + 1.07·13-s − 1.26i·17-s + 0.312i·19-s + (−0.703 + 0.710i)23-s − 0.200·25-s + 0.312·29-s − 1.10·31-s − 0.490·35-s + 1.49i·37-s + 1.03·41-s + 0.881i·43-s − 1.50·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.710 - 0.703i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.710 - 0.703i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.371186564\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.371186564\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - 2.23iT \) |
| 23 | \( 1 + (16.1 - 16.3i)T \) |
| good | 7 | \( 1 - 7.68iT - 49T^{2} \) |
| 11 | \( 1 + 8.06iT - 121T^{2} \) |
| 13 | \( 1 - 14.0T + 169T^{2} \) |
| 17 | \( 1 + 21.5iT - 289T^{2} \) |
| 19 | \( 1 - 5.94iT - 361T^{2} \) |
| 29 | \( 1 - 9.04T + 841T^{2} \) |
| 31 | \( 1 + 34.1T + 961T^{2} \) |
| 37 | \( 1 - 55.1iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 42.2T + 1.68e3T^{2} \) |
| 43 | \( 1 - 37.9iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 70.5T + 2.20e3T^{2} \) |
| 53 | \( 1 + 21.2iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 5.95T + 3.48e3T^{2} \) |
| 61 | \( 1 + 25.1iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 21.0iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 85.4T + 5.04e3T^{2} \) |
| 73 | \( 1 - 75.5T + 5.32e3T^{2} \) |
| 79 | \( 1 - 53.4iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 80.9iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 57.7iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 56.0iT - 9.40e3T^{2} \) |
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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
−8.350733857114258269168000830753, −8.041186251046701852882406952381, −6.94970721722299463081070034357, −6.24523945741271338611840087427, −5.65855780339884754344836351186, −4.94942526159782203868905645296, −3.74588932157234223539413866162, −3.10709710761329468120146694270, −2.25673731768440520547436546025, −1.13440299050434854714586806146,
0.29893517841896505297364307447, 1.35245240734774763858962211653, 2.19558213965323179084133332968, 3.69829999792122018162291001458, 4.00138851696813145284072497456, 4.86087836065352501124829452154, 5.84683495979168345381172020150, 6.49214501758589825496326048437, 7.32249553339050165989366221148, 7.941262391448024227941295461300
