L(s) = 1 | + (0.707 − 0.707i)3-s + i·5-s − 1.41i·7-s − 1.00i·9-s + (0.707 + 0.707i)15-s + (−1.00 − 1.00i)21-s + 1.41·23-s − 25-s + (−0.707 − 0.707i)27-s + 1.41·35-s + 2i·41-s + 1.41i·43-s + 1.00·45-s − 1.41·47-s − 1.00·49-s + ⋯ |
L(s) = 1 | + (0.707 − 0.707i)3-s + i·5-s − 1.41i·7-s − 1.00i·9-s + (0.707 + 0.707i)15-s + (−1.00 − 1.00i)21-s + 1.41·23-s − 25-s + (−0.707 − 0.707i)27-s + 1.41·35-s + 2i·41-s + 1.41i·43-s + 1.00·45-s − 1.41·47-s − 1.00·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.258232467\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.258232467\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.707 + 0.707i)T \) |
| 5 | \( 1 - iT \) |
good | 7 | \( 1 + 1.41iT - T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 - 1.41T + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - 2iT - T^{2} \) |
| 43 | \( 1 - 1.41iT - T^{2} \) |
| 47 | \( 1 + 1.41T + T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 + 1.41iT - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 + 1.41T + T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−10.05559221458814024568136367743, −9.407164531748578763769288639892, −8.176526296393001303838460109806, −7.55801903984939204258646581233, −6.84288064031452528286995467568, −6.28281240089725782717700820834, −4.65276370543786748256698173174, −3.53727090020670864417844174626, −2.83195966371016843776758153539, −1.34317531009062409967794383494,
1.91097938139596868916120632599, 2.97471217989071320800122459540, 4.13704963367706339077909785761, 5.18355593103578933558912311677, 5.60303087869201692278661580958, 7.08962373701661366401641086813, 8.248050013143878694153461153911, 8.802616603375437587928455014691, 9.234841641994241248913082588130, 10.09960419406987210799869225073