L(s) = 1 | + (−1.39 − 0.258i)2-s + (1.86 + 0.719i)4-s + 0.732i·5-s + (−2.40 − 1.48i)8-s + (0.189 − 1.01i)10-s − 2.03·11-s − 1.41·13-s + (2.96 + 2.68i)16-s − 4.19i·17-s + 5.56i·19-s + (−0.526 + 1.36i)20-s + (2.83 + 0.526i)22-s − 2.03·23-s + 4.46·25-s + (1.96 + 0.366i)26-s + ⋯ |
L(s) = 1 | + (−0.983 − 0.183i)2-s + (0.933 + 0.359i)4-s + 0.327i·5-s + (−0.851 − 0.524i)8-s + (0.0599 − 0.321i)10-s − 0.613·11-s − 0.392·13-s + (0.741 + 0.671i)16-s − 1.01i·17-s + 1.27i·19-s + (−0.117 + 0.305i)20-s + (0.603 + 0.112i)22-s − 0.424·23-s + 0.892·25-s + (0.385 + 0.0717i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.244 + 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.244 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7619195364\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7619195364\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.39 + 0.258i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - 0.732iT - 5T^{2} \) |
| 11 | \( 1 + 2.03T + 11T^{2} \) |
| 13 | \( 1 + 1.41T + 13T^{2} \) |
| 17 | \( 1 + 4.19iT - 17T^{2} \) |
| 19 | \( 1 - 5.56iT - 19T^{2} \) |
| 23 | \( 1 + 2.03T + 23T^{2} \) |
| 29 | \( 1 + 5.27iT - 29T^{2} \) |
| 31 | \( 1 + 9.63iT - 31T^{2} \) |
| 37 | \( 1 - 3.46T + 37T^{2} \) |
| 41 | \( 1 - 8.73iT - 41T^{2} \) |
| 43 | \( 1 + 7.86iT - 43T^{2} \) |
| 47 | \( 1 + 10.7T + 47T^{2} \) |
| 53 | \( 1 + 9.14iT - 53T^{2} \) |
| 59 | \( 1 - 4.98T + 59T^{2} \) |
| 61 | \( 1 - 9.41T + 61T^{2} \) |
| 67 | \( 1 + 2.87iT - 67T^{2} \) |
| 71 | \( 1 - 9.08T + 71T^{2} \) |
| 73 | \( 1 + 1.69T + 73T^{2} \) |
| 79 | \( 1 + 4.98iT - 79T^{2} \) |
| 83 | \( 1 - 13.6T + 83T^{2} \) |
| 89 | \( 1 + 8.73iT - 89T^{2} \) |
| 97 | \( 1 + 15.0T + 97T^{2} \) |
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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
−9.324922557606663401700959322993, −8.105078973264887126240070953776, −7.88145281263824047728002679707, −6.89682924440544743739755167865, −6.15415205535425673141840618604, −5.16690010024121857408747188735, −3.88112383103761778353589738906, −2.83094437632854040751700146883, −2.01767613453965291186470089472, −0.44588759378127148790559080637,
1.07353162861505643655201912527, 2.29998349022199382262608128622, 3.28545718551983934793259092305, 4.76679334053650676247008641701, 5.46198727419568614927654133849, 6.56247123527732140572476819806, 7.11872562930816287088104410760, 8.085972116630005580028282848259, 8.652171078433275767529270599233, 9.326399209367175340774668739526