| 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
−9.105649247526298326958662425467, −7.977841285918091775608117013707, −7.34419575442636145792452656693, −7.02465383466194100583001886979, −5.97484470316833652082774093304, −5.39353825254079657184767943632, −4.34529825805116316693479017422, −3.41925853848012302056931581932, −2.31276944182535807430154591613, −1.11081067279625761814160164642,
0.924148798671134049438002472075, 2.05463718206618108492390116588, 2.91402319002299401761978193048, 3.64364970046822881602504452650, 4.76936009118578504379172318905, 5.79083396445194440743900950660, 6.35575603586554770096973022640, 7.33349917697525726630604605920, 8.009810503442726225345796017870, 8.978126070004694091042386808430
