| L(s) = 1 | + 2.23i·5-s + 1.15i·7-s + 7.52i·11-s + 12.8·13-s − 10.1i·17-s + 14.3i·19-s + (−12.7 − 19.1i)23-s − 5.00·25-s − 12.8·29-s + 27.6·31-s − 2.57·35-s − 53.2i·37-s + 46.5·41-s + 4.51i·43-s + 15.7·47-s + ⋯ |
| L(s) = 1 | + 0.447i·5-s + 0.164i·7-s + 0.684i·11-s + 0.989·13-s − 0.596i·17-s + 0.753i·19-s + (−0.554 − 0.831i)23-s − 0.200·25-s − 0.443·29-s + 0.893·31-s − 0.0736·35-s − 1.43i·37-s + 1.13·41-s + 0.105i·43-s + 0.336·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.831 - 0.554i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.831 - 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.250052231\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.250052231\) |
| \(L(2)\) |
|
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 | \( 1 - 2.23iT \) |
| 23 | \( 1 + (12.7 + 19.1i)T \) |
| good | 7 | \( 1 - 1.15iT - 49T^{2} \) |
| 11 | \( 1 - 7.52iT - 121T^{2} \) |
| 13 | \( 1 - 12.8T + 169T^{2} \) |
| 17 | \( 1 + 10.1iT - 289T^{2} \) |
| 19 | \( 1 - 14.3iT - 361T^{2} \) |
| 29 | \( 1 + 12.8T + 841T^{2} \) |
| 31 | \( 1 - 27.6T + 961T^{2} \) |
| 37 | \( 1 + 53.2iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 46.5T + 1.68e3T^{2} \) |
| 43 | \( 1 - 4.51iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 15.7T + 2.20e3T^{2} \) |
| 53 | \( 1 + 51.3iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 107.T + 3.48e3T^{2} \) |
| 61 | \( 1 - 86.6iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 20.6iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 38.6T + 5.04e3T^{2} \) |
| 73 | \( 1 + 47.4T + 5.32e3T^{2} \) |
| 79 | \( 1 - 61.0iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 26.9iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 89.4iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 54.0iT - 9.40e3T^{2} \) |
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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
−8.357041233357306636770399433194, −7.49640853161523901029614392987, −6.93641984831636601303009072208, −6.03537036331984448478310048230, −5.54629722772144353074973060131, −4.36378716846180964339402075591, −3.84076541827748805777040903419, −2.74114071824374103089855439677, −1.98713980054802567204449645998, −0.75760350142199854416085676055,
0.66833761479835867092283857033, 1.50951827967572605943894371837, 2.70885247585866380125767589004, 3.66617982539211086432782685236, 4.29355624020563104281254879534, 5.28589784781009775501774483105, 5.97737555591594421925581289038, 6.59192484848154021616271525485, 7.58551638123129824435707859360, 8.250030206236066897428239378969
