| L(s) = 1 | + 2.63·2-s − 0.266·3-s + 4.95·4-s + 3.08·5-s − 0.703·6-s − 0.847·7-s + 7.80·8-s − 2.92·9-s + 8.12·10-s − 11-s − 1.32·12-s + 3.32·13-s − 2.23·14-s − 0.821·15-s + 10.6·16-s + 2.04·17-s − 7.72·18-s − 6.48·19-s + 15.2·20-s + 0.225·21-s − 2.63·22-s + 7.46·23-s − 2.08·24-s + 4.48·25-s + 8.78·26-s + 1.58·27-s − 4.20·28-s + ⋯ |
| L(s) = 1 | + 1.86·2-s − 0.153·3-s + 2.47·4-s + 1.37·5-s − 0.287·6-s − 0.320·7-s + 2.76·8-s − 0.976·9-s + 2.56·10-s − 0.301·11-s − 0.381·12-s + 0.923·13-s − 0.597·14-s − 0.212·15-s + 2.66·16-s + 0.496·17-s − 1.82·18-s − 1.48·19-s + 3.41·20-s + 0.0493·21-s − 0.562·22-s + 1.55·23-s − 0.424·24-s + 0.897·25-s + 1.72·26-s + 0.304·27-s − 0.794·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2563 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2563 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.824228941\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.824228941\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + T \) |
| 233 | \( 1 - T \) |
| good | 2 | \( 1 - 2.63T + 2T^{2} \) |
| 3 | \( 1 + 0.266T + 3T^{2} \) |
| 5 | \( 1 - 3.08T + 5T^{2} \) |
| 7 | \( 1 + 0.847T + 7T^{2} \) |
| 13 | \( 1 - 3.32T + 13T^{2} \) |
| 17 | \( 1 - 2.04T + 17T^{2} \) |
| 19 | \( 1 + 6.48T + 19T^{2} \) |
| 23 | \( 1 - 7.46T + 23T^{2} \) |
| 29 | \( 1 + 5.86T + 29T^{2} \) |
| 31 | \( 1 - 9.68T + 31T^{2} \) |
| 37 | \( 1 - 10.9T + 37T^{2} \) |
| 41 | \( 1 - 5.33T + 41T^{2} \) |
| 43 | \( 1 + 5.60T + 43T^{2} \) |
| 47 | \( 1 + 5.31T + 47T^{2} \) |
| 53 | \( 1 - 4.73T + 53T^{2} \) |
| 59 | \( 1 + 8.15T + 59T^{2} \) |
| 61 | \( 1 + 3.62T + 61T^{2} \) |
| 67 | \( 1 + 7.60T + 67T^{2} \) |
| 71 | \( 1 + 0.574T + 71T^{2} \) |
| 73 | \( 1 + 7.39T + 73T^{2} \) |
| 79 | \( 1 + 5.19T + 79T^{2} \) |
| 83 | \( 1 + 13.9T + 83T^{2} \) |
| 89 | \( 1 - 7.44T + 89T^{2} \) |
| 97 | \( 1 + 8.10T + 97T^{2} \) |
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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
−8.897248661498365036480271234439, −7.957930741584329858270292067169, −6.77077967037472767862376974642, −6.12232863402539423454844556205, −5.88974559486960254103597512265, −5.05246613399612458661496552900, −4.25622393711395865712010406527, −3.05270775261363102364549269903, −2.63565068697196372440989148033, −1.49309903698512035540604098087,
1.49309903698512035540604098087, 2.63565068697196372440989148033, 3.05270775261363102364549269903, 4.25622393711395865712010406527, 5.05246613399612458661496552900, 5.88974559486960254103597512265, 6.12232863402539423454844556205, 6.77077967037472767862376974642, 7.957930741584329858270292067169, 8.897248661498365036480271234439
