L(s) = 1 | + (−1.58 − 0.707i)3-s + 2.23i·5-s + 7-s + (2.00 + 2.23i)9-s − 2.23i·11-s − 4.24i·13-s + (1.58 − 3.53i)15-s + 2.23i·17-s + (−1 + 4.24i)19-s + (−1.58 − 0.707i)21-s + 4.47i·23-s + (−1.58 − 4.94i)27-s + 4.24i·31-s + (−1.58 + 3.53i)33-s + 2.23i·35-s + ⋯ |
L(s) = 1 | + (−0.912 − 0.408i)3-s + 0.999i·5-s + 0.377·7-s + (0.666 + 0.745i)9-s − 0.674i·11-s − 1.17i·13-s + (0.408 − 0.912i)15-s + 0.542i·17-s + (−0.229 + 0.973i)19-s + (−0.345 − 0.154i)21-s + 0.932i·23-s + (−0.304 − 0.952i)27-s + 0.762i·31-s + (−0.275 + 0.615i)33-s + 0.377i·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.606 - 0.794i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.606 - 0.794i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.984512 + 0.487030i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.984512 + 0.487030i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.58 + 0.707i)T \) |
| 19 | \( 1 + (1 - 4.24i)T \) |
good | 5 | \( 1 - 2.23iT - 5T^{2} \) |
| 7 | \( 1 - T + 7T^{2} \) |
| 11 | \( 1 + 2.23iT - 11T^{2} \) |
| 13 | \( 1 + 4.24iT - 13T^{2} \) |
| 17 | \( 1 - 2.23iT - 17T^{2} \) |
| 23 | \( 1 - 4.47iT - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 4.24iT - 31T^{2} \) |
| 37 | \( 1 - 4.24iT - 37T^{2} \) |
| 41 | \( 1 - 9.48T + 41T^{2} \) |
| 43 | \( 1 - 7T + 43T^{2} \) |
| 47 | \( 1 - 11.1iT - 47T^{2} \) |
| 53 | \( 1 + 9.48T + 53T^{2} \) |
| 59 | \( 1 - 9.48T + 59T^{2} \) |
| 61 | \( 1 - 11T + 61T^{2} \) |
| 67 | \( 1 - 8.48iT - 67T^{2} \) |
| 71 | \( 1 - 9.48T + 71T^{2} \) |
| 73 | \( 1 - 5T + 73T^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 + 8.94iT - 83T^{2} \) |
| 89 | \( 1 + 9.48T + 89T^{2} \) |
| 97 | \( 1 - 12.7iT - 97T^{2} \) |
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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
−10.59092633645048983370819682278, −9.632921044155280373860561905522, −8.169317183505829865204911218770, −7.71194210846944611068627894942, −6.66102578868293613389154649559, −5.94450039628884566817835354187, −5.22611103466608517846201326038, −3.86482989066362919673562776764, −2.73280541175669526862216654760, −1.23007196348289052772988160560,
0.68363830704138148030605529812, 2.15996640133711244822286571388, 4.11601989922552432960035114644, 4.65181933028358657893359928436, 5.37321178214208392813096354572, 6.52740168134791730591883487123, 7.24037430816403104534538481196, 8.487363595749384958201571980732, 9.298039000950643963906455288054, 9.807213296822868958228757255406