L(s) = 1 | − 3i·3-s − 6.85i·5-s + 7·7-s − 9·9-s + 30.4i·11-s + 45.1i·13-s − 20.5·15-s − 99.4·17-s − 33.9i·19-s − 21i·21-s + 108.·23-s + 78.0·25-s + 27i·27-s − 187. i·29-s − 174.·31-s + ⋯ |
L(s) = 1 | − 0.577i·3-s − 0.613i·5-s + 0.377·7-s − 0.333·9-s + 0.834i·11-s + 0.962i·13-s − 0.354·15-s − 1.41·17-s − 0.410i·19-s − 0.218i·21-s + 0.982·23-s + 0.624·25-s + 0.192i·27-s − 1.20i·29-s − 1.01·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.258 + 0.965i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.258 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.927805606\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.927805606\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + 3iT \) |
| 7 | \( 1 - 7T \) |
good | 5 | \( 1 + 6.85iT - 125T^{2} \) |
| 11 | \( 1 - 30.4iT - 1.33e3T^{2} \) |
| 13 | \( 1 - 45.1iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 99.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 33.9iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 108.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 187. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 174.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 241. iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 474.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 480. iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 516.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 179. iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 22.3iT - 2.05e5T^{2} \) |
| 61 | \( 1 + 932. iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 665. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 129.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.08e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 739.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 81.6iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 1.01e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 610.T + 9.12e5T^{2} \) |
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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
−8.993552056101652830425866950199, −8.393229869147668893649954289945, −7.22524229690237552307417125646, −6.88111374163034795061045688466, −5.74444824904087199906469289330, −4.72232126049197323620670616425, −4.17035251774853256981987277880, −2.53367083635642573388743200486, −1.75884235637082867770877245526, −0.56190256746990078347186960068,
0.873075394083658289056431995214, 2.47220700520306245847670880816, 3.26756290272134245391491393072, 4.24935026080331416227501788290, 5.25534350900019815195519385370, 5.99283100815000402222152495621, 6.98955897215727176342851246356, 7.79931939812896482603210880513, 8.811245141642482113655227877027, 9.216790416968112413947424332014