| L(s) = 1 | + (1.70 − 0.292i)3-s + i·5-s + 0.585i·7-s + (2.82 − i)9-s + 2.82·11-s + 2·13-s + (0.292 + 1.70i)15-s − 3.65i·17-s + 2.82i·19-s + (0.171 + i)21-s − 4.58·23-s − 25-s + (4.53 − 2.53i)27-s + 8i·29-s − 5.65i·31-s + ⋯ |
| L(s) = 1 | + (0.985 − 0.169i)3-s + 0.447i·5-s + 0.221i·7-s + (0.942 − 0.333i)9-s + 0.852·11-s + 0.554·13-s + (0.0756 + 0.440i)15-s − 0.886i·17-s + 0.648i·19-s + (0.0374 + 0.218i)21-s − 0.956·23-s − 0.200·25-s + (0.872 − 0.487i)27-s + 1.48i·29-s − 1.01i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 - 0.169i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.985 - 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.41749 + 0.205884i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.41749 + 0.205884i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.70 + 0.292i)T \) |
| 5 | \( 1 - iT \) |
| good | 7 | \( 1 - 0.585iT - 7T^{2} \) |
| 11 | \( 1 - 2.82T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + 3.65iT - 17T^{2} \) |
| 19 | \( 1 - 2.82iT - 19T^{2} \) |
| 23 | \( 1 + 4.58T + 23T^{2} \) |
| 29 | \( 1 - 8iT - 29T^{2} \) |
| 31 | \( 1 + 5.65iT - 31T^{2} \) |
| 37 | \( 1 - 11.6T + 37T^{2} \) |
| 41 | \( 1 - 2iT - 41T^{2} \) |
| 43 | \( 1 - 11.8iT - 43T^{2} \) |
| 47 | \( 1 + 2.24T + 47T^{2} \) |
| 53 | \( 1 + 7.65iT - 53T^{2} \) |
| 59 | \( 1 + 9.65T + 59T^{2} \) |
| 61 | \( 1 - 5.65T + 61T^{2} \) |
| 67 | \( 1 + 13.0iT - 67T^{2} \) |
| 71 | \( 1 + 6.82T + 71T^{2} \) |
| 73 | \( 1 - 4.34T + 73T^{2} \) |
| 79 | \( 1 + 12.4iT - 79T^{2} \) |
| 83 | \( 1 + 9.07T + 83T^{2} \) |
| 89 | \( 1 - 15.3iT - 89T^{2} \) |
| 97 | \( 1 + 14.9T + 97T^{2} \) |
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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
−9.693966848503032513932197080517, −9.370592252677815463754627972309, −8.304493598092229058913766770230, −7.67791625374048439321451067749, −6.70358510862911210088224607376, −5.97051709847828387592251343797, −4.49469931838036930045635179805, −3.59988102841603146407170485291, −2.66388554224525805395656689073, −1.43274033432367894065856734640,
1.28921250116194665075053671117, 2.51780175269678124865062626304, 3.90962579124419425007846334516, 4.27088844762796763341626304012, 5.71450596088105319919339095239, 6.68616512729359268528128417691, 7.68363748831008746085137086802, 8.460029795984135626163793059771, 9.060224245359798815254066630876, 9.868707191170367954269349599403
