| L(s) = 1 | − i·2-s − 4-s + 0.115·5-s + i·8-s − 0.115i·10-s − 3.52i·11-s − 2.01i·13-s + 16-s − 2.44·17-s − 5.10i·19-s − 0.115·20-s − 3.52·22-s + 5.73i·23-s − 4.98·25-s − 2.01·26-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.5·4-s + 0.0517·5-s + 0.353i·8-s − 0.0365i·10-s − 1.06i·11-s − 0.557i·13-s + 0.250·16-s − 0.594·17-s − 1.17i·19-s − 0.0258·20-s − 0.751·22-s + 1.19i·23-s − 0.997·25-s − 0.394·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.912 - 0.409i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.912 - 0.409i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6214943907\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6214943907\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 0.115T + 5T^{2} \) |
| 11 | \( 1 + 3.52iT - 11T^{2} \) |
| 13 | \( 1 + 2.01iT - 13T^{2} \) |
| 17 | \( 1 + 2.44T + 17T^{2} \) |
| 19 | \( 1 + 5.10iT - 19T^{2} \) |
| 23 | \( 1 - 5.73iT - 23T^{2} \) |
| 29 | \( 1 - 4.48iT - 29T^{2} \) |
| 31 | \( 1 + 1.03iT - 31T^{2} \) |
| 37 | \( 1 - 4.69T + 37T^{2} \) |
| 41 | \( 1 - 4.06T + 41T^{2} \) |
| 43 | \( 1 + 6.70T + 43T^{2} \) |
| 47 | \( 1 - 2.96T + 47T^{2} \) |
| 53 | \( 1 + 9.17iT - 53T^{2} \) |
| 59 | \( 1 + 10.3T + 59T^{2} \) |
| 61 | \( 1 + 2.73iT - 61T^{2} \) |
| 67 | \( 1 + 2.54T + 67T^{2} \) |
| 71 | \( 1 - 5.12iT - 71T^{2} \) |
| 73 | \( 1 + 7.60iT - 73T^{2} \) |
| 79 | \( 1 + 4.88T + 79T^{2} \) |
| 83 | \( 1 + 17.9T + 83T^{2} \) |
| 89 | \( 1 + 10.0T + 89T^{2} \) |
| 97 | \( 1 + 12.2iT - 97T^{2} \) |
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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
−8.556759352798128242979773849454, −7.84190038931451997631259974352, −6.93334311675381957201553942557, −5.93275833371272674428838410453, −5.28755551209647238842558650722, −4.31904648540516993552958422719, −3.38980125084457060792136080797, −2.68250372273759157404234093985, −1.47183870405788085431040459849, −0.20146211934910601375061546296,
1.59520395998368288538339011501, 2.65731072375584323172067560543, 4.17542678750740409099594985981, 4.38595101501199628580680570584, 5.58531940354279586302874124843, 6.27500374999039514053290011169, 6.98679418501827525173809966399, 7.75223065847756095452511684443, 8.378572408188877981932131086371, 9.277509892531643240297809982039
