| L(s) = 1 | − i·2-s + 4-s + 2i·7-s − 3i·8-s + 11-s − 2i·13-s + 2·14-s − 16-s + 2i·17-s + 6·19-s − i·22-s − 4i·23-s − 2·26-s + 2i·28-s − 6·29-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 0.5·4-s + 0.755i·7-s − 1.06i·8-s + 0.301·11-s − 0.554i·13-s + 0.534·14-s − 0.250·16-s + 0.485i·17-s + 1.37·19-s − 0.213i·22-s − 0.834i·23-s − 0.392·26-s + 0.377i·28-s − 1.11·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.322991081\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.322991081\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 + iT - 2T^{2} \) |
| 7 | \( 1 - 2iT - 7T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 - 2iT - 17T^{2} \) |
| 19 | \( 1 - 6T + 19T^{2} \) |
| 23 | \( 1 + 4iT - 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 - 6iT - 37T^{2} \) |
| 41 | \( 1 - 10T + 41T^{2} \) |
| 43 | \( 1 - 6iT - 43T^{2} \) |
| 47 | \( 1 + 8iT - 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 + 8iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 2iT - 73T^{2} \) |
| 79 | \( 1 - 10T + 79T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 2iT - 97T^{2} \) |
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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
−8.978126070004694091042386808430, −8.009810503442726225345796017870, −7.33349917697525726630604605920, −6.35575603586554770096973022640, −5.79083396445194440743900950660, −4.76936009118578504379172318905, −3.64364970046822881602504452650, −2.91402319002299401761978193048, −2.05463718206618108492390116588, −0.924148798671134049438002472075,
1.11081067279625761814160164642, 2.31276944182535807430154591613, 3.41925853848012302056931581932, 4.34529825805116316693479017422, 5.39353825254079657184767943632, 5.97484470316833652082774093304, 7.02465383466194100583001886979, 7.34419575442636145792452656693, 7.977841285918091775608117013707, 9.105649247526298326958662425467
