L(s) = 1 | − 5-s + 1.36i·7-s + 2.32·11-s + 6.17·13-s + 6.91·17-s − 0.609i·19-s + (4.08 + 2.50i)23-s + 25-s − 0.841i·29-s + 0.0981·31-s − 1.36i·35-s − 6.85i·37-s − 6.58i·41-s − 0.293i·43-s + 3.82i·47-s + ⋯ |
L(s) = 1 | − 0.447·5-s + 0.514i·7-s + 0.702·11-s + 1.71·13-s + 1.67·17-s − 0.139i·19-s + (0.852 + 0.523i)23-s + 0.200·25-s − 0.156i·29-s + 0.0176·31-s − 0.230i·35-s − 1.12i·37-s − 1.02i·41-s − 0.0446i·43-s + 0.557i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0646i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0646i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.517443701\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.517443701\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 23 | \( 1 + (-4.08 - 2.50i)T \) |
good | 7 | \( 1 - 1.36iT - 7T^{2} \) |
| 11 | \( 1 - 2.32T + 11T^{2} \) |
| 13 | \( 1 - 6.17T + 13T^{2} \) |
| 17 | \( 1 - 6.91T + 17T^{2} \) |
| 19 | \( 1 + 0.609iT - 19T^{2} \) |
| 29 | \( 1 + 0.841iT - 29T^{2} \) |
| 31 | \( 1 - 0.0981T + 31T^{2} \) |
| 37 | \( 1 + 6.85iT - 37T^{2} \) |
| 41 | \( 1 + 6.58iT - 41T^{2} \) |
| 43 | \( 1 + 0.293iT - 43T^{2} \) |
| 47 | \( 1 - 3.82iT - 47T^{2} \) |
| 53 | \( 1 - 1.36T + 53T^{2} \) |
| 59 | \( 1 - 14.1iT - 59T^{2} \) |
| 61 | \( 1 + 2.54iT - 61T^{2} \) |
| 67 | \( 1 + 13.3iT - 67T^{2} \) |
| 71 | \( 1 + 4.87iT - 71T^{2} \) |
| 73 | \( 1 - 3.41T + 73T^{2} \) |
| 79 | \( 1 + 2.39iT - 79T^{2} \) |
| 83 | \( 1 - 2.10T + 83T^{2} \) |
| 89 | \( 1 + 13.1T + 89T^{2} \) |
| 97 | \( 1 - 7.80iT - 97T^{2} \) |
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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
−7.75206500948964582000565968147, −7.24793767077458799013214576012, −6.32292873013087836039888948870, −5.77460425228865525179849124815, −5.16837963698793847402340678140, −4.03010119291472939314796648969, −3.61068747214337867205938940715, −2.83239218578612341046043972214, −1.58563840841336419320746446099, −0.841811740560332858064412689915,
0.937096635401324111936616228942, 1.38246873533019472556835211106, 2.89349779180342596719434972907, 3.59446450564592872853454683618, 4.05746802743592887421946027378, 5.00017839114575831499840533952, 5.77834297153667063318596522502, 6.51067028443928906027973075360, 7.03872469345195955535909069014, 7.909705012178947117228393437423