| L(s) = 1 | − 8.36i·5-s − 2.43i·7-s + 2.41·11-s − 20.8i·13-s − 13.1·17-s + 32.7·19-s − 33.8i·23-s − 45.0·25-s + 34.7i·29-s − 33.7i·31-s − 20.4·35-s − 30.8i·37-s + 35.1·41-s + 27.9·43-s + 21.9i·47-s + ⋯ |
| L(s) = 1 | − 1.67i·5-s − 0.348i·7-s + 0.219·11-s − 1.60i·13-s − 0.775·17-s + 1.72·19-s − 1.47i·23-s − 1.80·25-s + 1.19i·29-s − 1.08i·31-s − 0.583·35-s − 0.833i·37-s + 0.857·41-s + 0.649·43-s + 0.467i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 + 0.258i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.965 + 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.688395675\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.688395675\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 8.36iT - 25T^{2} \) |
| 7 | \( 1 + 2.43iT - 49T^{2} \) |
| 11 | \( 1 - 2.41T + 121T^{2} \) |
| 13 | \( 1 + 20.8iT - 169T^{2} \) |
| 17 | \( 1 + 13.1T + 289T^{2} \) |
| 19 | \( 1 - 32.7T + 361T^{2} \) |
| 23 | \( 1 + 33.8iT - 529T^{2} \) |
| 29 | \( 1 - 34.7iT - 841T^{2} \) |
| 31 | \( 1 + 33.7iT - 961T^{2} \) |
| 37 | \( 1 + 30.8iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 35.1T + 1.68e3T^{2} \) |
| 43 | \( 1 - 27.9T + 1.84e3T^{2} \) |
| 47 | \( 1 - 21.9iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 26.3iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 93.3T + 3.48e3T^{2} \) |
| 61 | \( 1 + 27.7iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 9.97T + 4.48e3T^{2} \) |
| 71 | \( 1 + 64.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 45.5T + 5.32e3T^{2} \) |
| 79 | \( 1 - 82.8iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 59.4T + 6.88e3T^{2} \) |
| 89 | \( 1 + 139.T + 7.92e3T^{2} \) |
| 97 | \( 1 - 150.T + 9.40e3T^{2} \) |
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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
−8.954165949176458491217424272003, −7.933625526839202070258787809577, −7.50111094034963411670146685459, −6.16422843976343250798120220195, −5.37588388863407470082602723534, −4.73467800747084105996788941267, −3.85798956069382450776797280648, −2.65085370093428803966547081304, −1.16733195920597417685897709848, −0.49225840686527065640907329717,
1.62713142439393031249718112998, 2.65484497093404650396267440707, 3.46724982479596344731165872297, 4.41038282971980563567649917642, 5.65178447430282542018653760890, 6.41053568813474680195795372596, 7.13621947363342135118582121346, 7.59851331026563568944896110596, 8.891119965047269554454090028634, 9.539994999795262136804278728574
