| L(s) = 1 | − 1.41i·2-s − 2.00·4-s + 3.04i·5-s + 8.22·7-s + 2.82i·8-s + 4.30·10-s + 6.55i·11-s − 18.5·13-s − 11.6i·14-s + 4.00·16-s − 13.1i·17-s − 34.8·19-s − 6.08i·20-s + 9.26·22-s + 3.31i·23-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.500·4-s + 0.608i·5-s + 1.17·7-s + 0.353i·8-s + 0.430·10-s + 0.595i·11-s − 1.42·13-s − 0.831i·14-s + 0.250·16-s − 0.776i·17-s − 1.83·19-s − 0.304i·20-s + 0.421·22-s + 0.144i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1458 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1458 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.4321758943\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4321758943\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 1.41iT \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 3.04iT - 25T^{2} \) |
| 7 | \( 1 - 8.22T + 49T^{2} \) |
| 11 | \( 1 - 6.55iT - 121T^{2} \) |
| 13 | \( 1 + 18.5T + 169T^{2} \) |
| 17 | \( 1 + 13.1iT - 289T^{2} \) |
| 19 | \( 1 + 34.8T + 361T^{2} \) |
| 23 | \( 1 - 3.31iT - 529T^{2} \) |
| 29 | \( 1 - 28.4iT - 841T^{2} \) |
| 31 | \( 1 + 13.8T + 961T^{2} \) |
| 37 | \( 1 - 6.66T + 1.36e3T^{2} \) |
| 41 | \( 1 + 56.9iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 32.7T + 1.84e3T^{2} \) |
| 47 | \( 1 + 53.6iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 98.5iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 102. iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 7.18T + 3.72e3T^{2} \) |
| 67 | \( 1 - 30.0T + 4.48e3T^{2} \) |
| 71 | \( 1 - 9.77iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 129.T + 5.32e3T^{2} \) |
| 79 | \( 1 - 102.T + 6.24e3T^{2} \) |
| 83 | \( 1 + 108. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 23.2iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 51.2T + 9.40e3T^{2} \) |
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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
−8.989304185714563869618022179404, −8.229510572182436498316999790566, −7.30136736667982916161940021565, −6.69143770122737018953986153416, −5.12945141888869012262599643407, −4.81914291119925388077321261090, −3.67776485987459971582688250450, −2.43101580577122059586867138338, −1.85386503187915005560199026917, −0.11649204139320019006131753565,
1.36414661276739584507652065634, 2.60845675706147207467311722580, 4.31264985553288243098981225822, 4.61898482845015389362614061402, 5.62542956864327965995487684870, 6.41063733203571713624297772311, 7.46880362443392254845553860167, 8.189772059903430396137632117422, 8.632002833616108250148537362191, 9.518092753668421004187503305285
