L(s) = 1 | + (2.12 − 0.707i)5-s + 5.65i·11-s + (3 + 3i)13-s − 4i·19-s + (−2.82 + 2.82i)23-s + (3.99 − 3i)25-s + 1.41·29-s + 8·31-s + (7 − 7i)37-s + 1.41i·41-s + (−4 − 4i)43-s + (−2.82 − 2.82i)47-s + 7i·49-s + (−8.48 + 8.48i)53-s + (4.00 + 12i)55-s + ⋯ |
L(s) = 1 | + (0.948 − 0.316i)5-s + 1.70i·11-s + (0.832 + 0.832i)13-s − 0.917i·19-s + (−0.589 + 0.589i)23-s + (0.799 − 0.600i)25-s + 0.262·29-s + 1.43·31-s + (1.15 − 1.15i)37-s + 0.220i·41-s + (−0.609 − 0.609i)43-s + (−0.412 − 0.412i)47-s + i·49-s + (−1.16 + 1.16i)53-s + (0.539 + 1.61i)55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.927 - 0.374i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.927 - 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.80151 + 0.349883i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.80151 + 0.349883i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-2.12 + 0.707i)T \) |
good | 7 | \( 1 - 7iT^{2} \) |
| 11 | \( 1 - 5.65iT - 11T^{2} \) |
| 13 | \( 1 + (-3 - 3i)T + 13iT^{2} \) |
| 17 | \( 1 + 17iT^{2} \) |
| 19 | \( 1 + 4iT - 19T^{2} \) |
| 23 | \( 1 + (2.82 - 2.82i)T - 23iT^{2} \) |
| 29 | \( 1 - 1.41T + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 + (-7 + 7i)T - 37iT^{2} \) |
| 41 | \( 1 - 1.41iT - 41T^{2} \) |
| 43 | \( 1 + (4 + 4i)T + 43iT^{2} \) |
| 47 | \( 1 + (2.82 + 2.82i)T + 47iT^{2} \) |
| 53 | \( 1 + (8.48 - 8.48i)T - 53iT^{2} \) |
| 59 | \( 1 - 11.3T + 59T^{2} \) |
| 61 | \( 1 + 12T + 61T^{2} \) |
| 67 | \( 1 + (-8 + 8i)T - 67iT^{2} \) |
| 71 | \( 1 - 5.65iT - 71T^{2} \) |
| 73 | \( 1 + (3 + 3i)T + 73iT^{2} \) |
| 79 | \( 1 - 8iT - 79T^{2} \) |
| 83 | \( 1 + (-11.3 + 11.3i)T - 83iT^{2} \) |
| 89 | \( 1 + 7.07T + 89T^{2} \) |
| 97 | \( 1 + (-5 + 5i)T - 97iT^{2} \) |
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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
−10.27833062418303229340814878701, −9.554986572478026411359306535630, −9.010676387431366699934051139983, −7.85717995909130749594361805717, −6.82222045854486761116459844641, −6.12101403836104615228852874477, −4.93467161360282429426482441514, −4.20673475924094515768673189238, −2.52768408247980669586043311809, −1.50831075381735665946419007886,
1.12472130050722057994129818623, 2.72926743814641484300790816648, 3.60634290816935570638909679129, 5.12105931619814251721296513593, 6.13938816081715307555468413483, 6.39192041073941538223742684812, 8.129206882851449703019918657200, 8.405524314218624711007083207702, 9.669395142287701876671383313332, 10.34329704630875686323673539013