| L(s) = 1 | − 1.47i·3-s + i·5-s + 7-s + 0.818·9-s + 5.66i·11-s − 0.510i·13-s + 1.47·15-s + 0.408·17-s − 6.68i·19-s − 1.47i·21-s − 0.912·23-s − 25-s − 5.63i·27-s + 8.21i·29-s + 3.62·31-s + ⋯ |
| L(s) = 1 | − 0.852i·3-s + 0.447i·5-s + 0.377·7-s + 0.272·9-s + 1.70i·11-s − 0.141i·13-s + 0.381·15-s + 0.0991·17-s − 1.53i·19-s − 0.322i·21-s − 0.190·23-s − 0.200·25-s − 1.08i·27-s + 1.52i·29-s + 0.650·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 - 0.258i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.965 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.901153197\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.901153197\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - iT \) |
| 7 | \( 1 - T \) |
| good | 3 | \( 1 + 1.47iT - 3T^{2} \) |
| 11 | \( 1 - 5.66iT - 11T^{2} \) |
| 13 | \( 1 + 0.510iT - 13T^{2} \) |
| 17 | \( 1 - 0.408T + 17T^{2} \) |
| 19 | \( 1 + 6.68iT - 19T^{2} \) |
| 23 | \( 1 + 0.912T + 23T^{2} \) |
| 29 | \( 1 - 8.21iT - 29T^{2} \) |
| 31 | \( 1 - 3.62T + 31T^{2} \) |
| 37 | \( 1 - 7.33iT - 37T^{2} \) |
| 41 | \( 1 - 11.2T + 41T^{2} \) |
| 43 | \( 1 - 8.39iT - 43T^{2} \) |
| 47 | \( 1 + 11.4T + 47T^{2} \) |
| 53 | \( 1 - 12.8iT - 53T^{2} \) |
| 59 | \( 1 - 8.85iT - 59T^{2} \) |
| 61 | \( 1 - 2.35iT - 61T^{2} \) |
| 67 | \( 1 + 14.9iT - 67T^{2} \) |
| 71 | \( 1 - 13.3T + 71T^{2} \) |
| 73 | \( 1 - 0.470T + 73T^{2} \) |
| 79 | \( 1 - 10.1T + 79T^{2} \) |
| 83 | \( 1 + 7.14iT - 83T^{2} \) |
| 89 | \( 1 - 2.91T + 89T^{2} \) |
| 97 | \( 1 - 10.5T + 97T^{2} \) |
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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
−9.137052182335498431378368709765, −7.991595099795385130821351156553, −7.45801752789047430409483723056, −6.85123939341063444150440246671, −6.25304352010746482254279324940, −4.86358649935975083538648186388, −4.48360129400047618872100915973, −3.00974490852470063465565099287, −2.12749777398014682386458597823, −1.17521697948230660643125225587,
0.76181494096886640658998991092, 2.12048253998005983822448742132, 3.59631830752215001414265626236, 3.95405340649440897916168296568, 5.03608561751065335249823632084, 5.72261942871192627787827143125, 6.46056361133090226377561709994, 7.80530397425988605851079539725, 8.219863247805015687527044120840, 9.045648990081996610761173031480
