L(s) = 1 | + (−1.26 + 4.83i)5-s − 6.26i·7-s + 1.03i·11-s + 3i·13-s − 15.7·17-s − 18.7·19-s + 34.6·23-s + (−21.7 − 12.2i)25-s − 3.10i·29-s + 17.7·31-s + (30.2 + 7.93i)35-s − 43.3i·37-s + 46.2i·41-s − 20.7i·43-s + 78.8·47-s + ⋯ |
L(s) = 1 | + (−0.253 + 0.967i)5-s − 0.894i·7-s + 0.0939i·11-s + 0.230i·13-s − 0.928·17-s − 0.988·19-s + 1.50·23-s + (−0.871 − 0.490i)25-s − 0.106i·29-s + 0.573·31-s + (0.865 + 0.226i)35-s − 1.17i·37-s + 1.12i·41-s − 0.482i·43-s + 1.67·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 + 0.253i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.967 + 0.253i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.609332088\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.609332088\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (1.26 - 4.83i)T \) |
good | 7 | \( 1 + 6.26iT - 49T^{2} \) |
| 11 | \( 1 - 1.03iT - 121T^{2} \) |
| 13 | \( 1 - 3iT - 169T^{2} \) |
| 17 | \( 1 + 15.7T + 289T^{2} \) |
| 19 | \( 1 + 18.7T + 361T^{2} \) |
| 23 | \( 1 - 34.6T + 529T^{2} \) |
| 29 | \( 1 + 3.10iT - 841T^{2} \) |
| 31 | \( 1 - 17.7T + 961T^{2} \) |
| 37 | \( 1 + 43.3iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 46.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 20.7iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 78.8T + 2.20e3T^{2} \) |
| 53 | \( 1 + 44.2T + 2.80e3T^{2} \) |
| 59 | \( 1 + 90.5iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 8.78T + 3.72e3T^{2} \) |
| 67 | \( 1 + 19.3iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 56.6iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 109. iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 39T + 6.24e3T^{2} \) |
| 83 | \( 1 - 75.7T + 6.88e3T^{2} \) |
| 89 | \( 1 + 90.5iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 8.88iT - 9.40e3T^{2} \) |
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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
−8.843577666676201016524482412966, −8.005996719578144528257159481861, −7.05699063204036262417243883318, −6.82419301904261006438008456575, −5.86151415439831022292134878563, −4.59935823275189554004310515378, −4.01247667522379546447117761314, −3.00846635540368748784201598221, −2.05760295007501965512275015863, −0.56334671977084237358535064952,
0.76774650437747647889858190534, 2.02026456487708909315387636632, 2.99883000136302208247661395016, 4.22909758676298651061515978851, 4.88786862500270815746622905587, 5.69135479484967726971697374314, 6.50442963044850219516910809944, 7.45583456059917773907532619839, 8.421143869370432805675187648329, 8.855676224145136528296658449352