L(s) = 1 | + (2.07 + 3.58i)5-s + (−2.19 − 1.48i)7-s + (−0.434 + 0.752i)11-s + (2.86 − 4.95i)13-s + (1.44 + 2.50i)17-s + (2.00 − 3.47i)19-s + (2.91 + 5.04i)23-s + (−6.09 + 10.5i)25-s + (0.900 + 1.55i)29-s + 2.96·31-s + (0.789 − 10.9i)35-s + (−2.64 + 4.58i)37-s + (5.89 − 10.2i)41-s + (2.00 + 3.47i)43-s + 2.34·47-s + ⋯ |
L(s) = 1 | + (0.926 + 1.60i)5-s + (−0.827 − 0.561i)7-s + (−0.130 + 0.226i)11-s + (0.793 − 1.37i)13-s + (0.350 + 0.607i)17-s + (0.460 − 0.797i)19-s + (0.607 + 1.05i)23-s + (−1.21 + 2.11i)25-s + (0.167 + 0.289i)29-s + 0.531·31-s + (0.133 − 1.84i)35-s + (−0.435 + 0.754i)37-s + (0.920 − 1.59i)41-s + (0.306 + 0.530i)43-s + 0.341·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.462 - 0.886i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.462 - 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.085286075\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.085286075\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (2.19 + 1.48i)T \) |
good | 5 | \( 1 + (-2.07 - 3.58i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (0.434 - 0.752i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.86 + 4.95i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-1.44 - 2.50i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.00 + 3.47i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.91 - 5.04i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.900 - 1.55i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 2.96T + 31T^{2} \) |
| 37 | \( 1 + (2.64 - 4.58i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-5.89 + 10.2i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-2.00 - 3.47i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 2.34T + 47T^{2} \) |
| 53 | \( 1 + (-1.09 - 1.88i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + 3.05T + 59T^{2} \) |
| 61 | \( 1 - 5.63T + 61T^{2} \) |
| 67 | \( 1 - 2.51T + 67T^{2} \) |
| 71 | \( 1 + 1.09T + 71T^{2} \) |
| 73 | \( 1 + (-0.723 - 1.25i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 - 2.12T + 79T^{2} \) |
| 83 | \( 1 + (-2.18 - 3.78i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (5.83 - 10.1i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (3.98 + 6.90i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.047330335458629299101835805153, −7.87308596501458826899192538299, −7.22150937192847046963413894644, −6.61669102952469228999165751888, −5.90593903475832977147223587394, −5.30925325625260844937843966156, −3.77314242157514983289665739107, −3.17928071942590431252757876133, −2.52948103743496200122152675865, −1.10993047245377566872401595585,
0.76443006208061171198409506246, 1.77001930015756686888652142719, 2.77775356762537130410335084203, 4.01349765829263782478424947861, 4.77357414959955223885799434646, 5.63503647164046666716957968902, 6.11992444628243660992024761998, 6.90721944809883552666036873444, 8.172709752524934269995646737651, 8.709475403205754628601435543444