| L(s) = 1 | + 3.23i·3-s + 3.23i·5-s − 7-s − 7.47·9-s − 4i·11-s + 3.23i·13-s − 10.4·15-s − 2·17-s − 3.23i·19-s − 3.23i·21-s + 2.47·23-s − 5.47·25-s − 14.4i·27-s + 10.4i·29-s + 12.9·33-s + ⋯ |
| L(s) = 1 | + 1.86i·3-s + 1.44i·5-s − 0.377·7-s − 2.49·9-s − 1.20i·11-s + 0.897i·13-s − 2.70·15-s − 0.485·17-s − 0.742i·19-s − 0.706i·21-s + 0.515·23-s − 1.09·25-s − 2.78i·27-s + 1.94i·29-s + 2.25·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.366880 - 0.885728i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.366880 - 0.885728i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 3 | \( 1 - 3.23iT - 3T^{2} \) |
| 5 | \( 1 - 3.23iT - 5T^{2} \) |
| 11 | \( 1 + 4iT - 11T^{2} \) |
| 13 | \( 1 - 3.23iT - 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 + 3.23iT - 19T^{2} \) |
| 23 | \( 1 - 2.47T + 23T^{2} \) |
| 29 | \( 1 - 10.4iT - 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 2.47iT - 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - 8.94iT - 43T^{2} \) |
| 47 | \( 1 - 4.94T + 47T^{2} \) |
| 53 | \( 1 + 8.94iT - 53T^{2} \) |
| 59 | \( 1 + 4.76iT - 59T^{2} \) |
| 61 | \( 1 + 3.23iT - 61T^{2} \) |
| 67 | \( 1 - 10.4iT - 67T^{2} \) |
| 71 | \( 1 + 12.9T + 71T^{2} \) |
| 73 | \( 1 - 2.94T + 73T^{2} \) |
| 79 | \( 1 + 12.9T + 79T^{2} \) |
| 83 | \( 1 - 4.76iT - 83T^{2} \) |
| 89 | \( 1 + 2T + 89T^{2} \) |
| 97 | \( 1 + 6.94T + 97T^{2} \) |
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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
−10.74060109614151104522737336331, −9.892801630386205070622239854642, −9.122796323526836254069095273004, −8.519615043385075049561824926924, −7.01796968015891547176865444346, −6.29074335555247827639198176574, −5.29451699009571156081318835913, −4.25789921726908254593588580305, −3.29617467420183262889697993053, −2.81201801944003636703451199460,
0.45196192183292289932409834294, 1.57908861770238165148503998755, 2.58639979338166996416655223287, 4.26578971009698471628741547545, 5.46612994775075670246656689469, 6.11425895437490720080515514885, 7.24204173507157306222795375533, 7.77235687113535289865414633507, 8.587634974233567668442038185345, 9.300949879744494685952313412585
