L(s) = 1 | + i·2-s − 4-s + 5-s − i·8-s + i·10-s + 1.37i·11-s − 4.02i·13-s + 16-s − 0.648·17-s + 5.77i·19-s − 20-s − 1.37·22-s − 1.66i·23-s + 25-s + 4.02·26-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.5·4-s + 0.447·5-s − 0.353i·8-s + 0.316i·10-s + 0.415i·11-s − 1.11i·13-s + 0.250·16-s − 0.157·17-s + 1.32i·19-s − 0.223·20-s − 0.293·22-s − 0.347i·23-s + 0.200·25-s + 0.789·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.508 - 0.860i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.508 - 0.860i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.905446120\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.905446120\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 - 1.37iT - 11T^{2} \) |
| 13 | \( 1 + 4.02iT - 13T^{2} \) |
| 17 | \( 1 + 0.648T + 17T^{2} \) |
| 19 | \( 1 - 5.77iT - 19T^{2} \) |
| 23 | \( 1 + 1.66iT - 23T^{2} \) |
| 29 | \( 1 - 2.53iT - 29T^{2} \) |
| 31 | \( 1 + 6.20iT - 31T^{2} \) |
| 37 | \( 1 - 10.9T + 37T^{2} \) |
| 41 | \( 1 - 6.26T + 41T^{2} \) |
| 43 | \( 1 + 7.21T + 43T^{2} \) |
| 47 | \( 1 + 0.598T + 47T^{2} \) |
| 53 | \( 1 - 7.48iT - 53T^{2} \) |
| 59 | \( 1 - 3.71T + 59T^{2} \) |
| 61 | \( 1 + 4.94iT - 61T^{2} \) |
| 67 | \( 1 + 3.76T + 67T^{2} \) |
| 71 | \( 1 - 3.97iT - 71T^{2} \) |
| 73 | \( 1 - 7.53iT - 73T^{2} \) |
| 79 | \( 1 - 10.4T + 79T^{2} \) |
| 83 | \( 1 - 4.08T + 83T^{2} \) |
| 89 | \( 1 - 9.76T + 89T^{2} \) |
| 97 | \( 1 + 7.65iT - 97T^{2} \) |
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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
−8.210421658098373952530449844261, −7.84225862280366398491363404125, −7.05922824903284456316947129951, −6.08878974574343900770729620420, −5.79700712849021719544238194921, −4.87220435642699329531745042737, −4.10239315708518658402141826860, −3.13094565643052889993189194736, −2.09715010833598922238676932033, −0.829619874333495730260006858687,
0.74100620111524828707310295158, 1.87363805689378556160022972578, 2.65145298822757771154554395375, 3.54786218011773583150660118804, 4.50167457975838979220554191115, 5.05193708978245009136668511502, 6.08534494109551856907322489170, 6.68529200638469098944654714747, 7.55502204425418166073699230419, 8.466609390409410689985347357086