L(s) = 1 | + (2 − i)5-s + 2i·7-s + 2·11-s − 2i·13-s + 6i·17-s + 8·19-s + 4i·23-s + (3 − 4i)25-s − 8·29-s + (2 + 4i)35-s − 10i·37-s − 2·41-s + 12i·43-s + 3·49-s + 10i·53-s + ⋯ |
L(s) = 1 | + (0.894 − 0.447i)5-s + 0.755i·7-s + 0.603·11-s − 0.554i·13-s + 1.45i·17-s + 1.83·19-s + 0.834i·23-s + (0.600 − 0.800i)25-s − 1.48·29-s + (0.338 + 0.676i)35-s − 1.64i·37-s − 0.312·41-s + 1.82i·43-s + 0.428·49-s + 1.37i·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.362698738\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.362698738\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-2 + i)T \) |
good | 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 - 6iT - 17T^{2} \) |
| 19 | \( 1 - 8T + 19T^{2} \) |
| 23 | \( 1 - 4iT - 23T^{2} \) |
| 29 | \( 1 + 8T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 10iT - 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 - 12iT - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 - 10iT - 53T^{2} \) |
| 59 | \( 1 - 6T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 8iT - 67T^{2} \) |
| 71 | \( 1 - 4T + 71T^{2} \) |
| 73 | \( 1 - 4iT - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 - 4iT - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 8iT - 97T^{2} \) |
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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
−9.158011750354362006742613272191, −8.082861417968255847682690023734, −7.43578674160238268802876432587, −6.30820571926707634797879843440, −5.66002179569545586628642043882, −5.29069678724686815112295690940, −4.04366971032809460292697041116, −3.15199237533902869237158795073, −2.02339348374356193843564255193, −1.18449693014667912075554981992,
0.868132391018211758700877080784, 1.98278026489574169808214591022, 3.05015168026851837114264694731, 3.85402845234041621627032617068, 4.98839706083225472542326402310, 5.54506790601799216868597445911, 6.76145677851338097963760276451, 6.94788724628019218087708838317, 7.80054338687082878417070857827, 8.980020917645437582523382568979