| L(s) = 1 | + (−1.38 − 1.15i)2-s + (−0.824 − 4.67i)4-s + (3.13 − 17.7i)5-s + (−14.1 − 24.4i)7-s + (−11.4 + 19.9i)8-s + (−24.9 + 20.9i)10-s + (1.89 − 3.28i)11-s + (44.0 + 16.0i)13-s + (−8.84 + 50.1i)14-s + (3.23 − 1.17i)16-s + (−14.5 − 12.1i)17-s + (75.4 + 34.0i)19-s − 85.7·20-s + (−6.42 + 2.33i)22-s + (2.85 + 16.1i)23-s + ⋯ |
| L(s) = 1 | + (−0.488 − 0.409i)2-s + (−0.103 − 0.584i)4-s + (0.280 − 1.59i)5-s + (−0.762 − 1.32i)7-s + (−0.507 + 0.879i)8-s + (−0.788 + 0.661i)10-s + (0.0519 − 0.0900i)11-s + (0.939 + 0.342i)13-s + (−0.168 + 0.957i)14-s + (0.0504 − 0.0183i)16-s + (−0.207 − 0.173i)17-s + (0.911 + 0.411i)19-s − 0.958·20-s + (−0.0622 + 0.0226i)22-s + (0.0258 + 0.146i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.917 - 0.397i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.917 - 0.397i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.185509 + 0.895133i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.185509 + 0.895133i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 19 | \( 1 + (-75.4 - 34.0i)T \) |
| good | 2 | \( 1 + (1.38 + 1.15i)T + (1.38 + 7.87i)T^{2} \) |
| 5 | \( 1 + (-3.13 + 17.7i)T + (-117. - 42.7i)T^{2} \) |
| 7 | \( 1 + (14.1 + 24.4i)T + (-171.5 + 297. i)T^{2} \) |
| 11 | \( 1 + (-1.89 + 3.28i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-44.0 - 16.0i)T + (1.68e3 + 1.41e3i)T^{2} \) |
| 17 | \( 1 + (14.5 + 12.1i)T + (853. + 4.83e3i)T^{2} \) |
| 23 | \( 1 + (-2.85 - 16.1i)T + (-1.14e4 + 4.16e3i)T^{2} \) |
| 29 | \( 1 + (108. - 90.8i)T + (4.23e3 - 2.40e4i)T^{2} \) |
| 31 | \( 1 + (89.1 + 154. i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + 29.5T + 5.06e4T^{2} \) |
| 41 | \( 1 + (-328. + 119. i)T + (5.27e4 - 4.43e4i)T^{2} \) |
| 43 | \( 1 + (13.5 - 77.1i)T + (-7.47e4 - 2.71e4i)T^{2} \) |
| 47 | \( 1 + (158. - 132. i)T + (1.80e4 - 1.02e5i)T^{2} \) |
| 53 | \( 1 + (67.7 + 384. i)T + (-1.39e5 + 5.09e4i)T^{2} \) |
| 59 | \( 1 + (26.7 + 22.4i)T + (3.56e4 + 2.02e5i)T^{2} \) |
| 61 | \( 1 + (117. + 667. i)T + (-2.13e5 + 7.76e4i)T^{2} \) |
| 67 | \( 1 + (579. - 486. i)T + (5.22e4 - 2.96e5i)T^{2} \) |
| 71 | \( 1 + (18.2 - 103. i)T + (-3.36e5 - 1.22e5i)T^{2} \) |
| 73 | \( 1 + (-803. + 292. i)T + (2.98e5 - 2.50e5i)T^{2} \) |
| 79 | \( 1 + (-591. + 215. i)T + (3.77e5 - 3.16e5i)T^{2} \) |
| 83 | \( 1 + (385. + 668. i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 + (-972. - 353. i)T + (5.40e5 + 4.53e5i)T^{2} \) |
| 97 | \( 1 + (-561. - 470. i)T + (1.58e5 + 8.98e5i)T^{2} \) |
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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
−11.57335512423483832273382608623, −10.59932833445174043957511628036, −9.531334251272209706648551558749, −9.098240724288287882407265663113, −7.81250375412609072328520405835, −6.24829262667341309499129515215, −5.13160712439367111522942160719, −3.83411592948984838202746067656, −1.48850685840266958263309345535, −0.51985744616162065823498887201,
2.66400821534241133027202352035, 3.49964092912246093676772807209, 5.85073990019729908259862773053, 6.60968908147333045588949574987, 7.60893015942638632119226052647, 8.868965273132849783392263396419, 9.624642784342216157275263854315, 10.80283133665404168792106286322, 11.83137185072585862885563527178, 12.84667363303409974698770635648
