L(s) = 1 | + 3i·3-s + 15.4·5-s + (6.29 + 17.4i)7-s − 9·9-s − 16.8·11-s + 9.10·13-s + 46.2i·15-s + 73.9i·17-s − 151. i·19-s + (−52.2 + 18.8i)21-s − 0.264i·23-s + 112.·25-s − 27i·27-s + 279. i·29-s + 147.·31-s + ⋯ |
L(s) = 1 | + 0.577i·3-s + 1.37·5-s + (0.339 + 0.940i)7-s − 0.333·9-s − 0.462·11-s + 0.194·13-s + 0.795i·15-s + 1.05i·17-s − 1.82i·19-s + (−0.543 + 0.196i)21-s − 0.00239i·23-s + 0.897·25-s − 0.192i·27-s + 1.79i·29-s + 0.856·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.571 - 0.820i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.571 - 0.820i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.419577305\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.419577305\) |
\(L(\frac{5}{2})\) |
|
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 | \( 1 + (-6.29 - 17.4i)T \) |
good | 5 | \( 1 - 15.4T + 125T^{2} \) |
| 11 | \( 1 + 16.8T + 1.33e3T^{2} \) |
| 13 | \( 1 - 9.10T + 2.19e3T^{2} \) |
| 17 | \( 1 - 73.9iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 151. iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 0.264iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 279. iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 147.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 418. iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 164. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 266.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 277.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 276. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 212. iT - 2.05e5T^{2} \) |
| 61 | \( 1 - 174.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 317.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 106. iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 755. iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 194. iT - 4.93e5T^{2} \) |
| 83 | \( 1 - 997. iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 988. iT - 7.04e5T^{2} \) |
| 97 | \( 1 - 1.02e3iT - 9.12e5T^{2} \) |
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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
−9.628080508993485317787850709307, −8.695413314690015989321714572542, −8.332798115259868366543079718827, −6.77705546216509432672875278433, −6.16738849931321491973863332055, −5.16010860778606633184361668600, −4.85006943087924691129695804424, −3.21815262748486084644097658134, −2.41277280569455083736521201063, −1.39604197912654797862707579359,
0.51878025386623944697069647341, 1.63173103037129866357096781804, 2.40839751401485752801952082958, 3.70054233536192471683442380076, 4.87359212757186483537546982182, 5.80357836894868910262419038721, 6.35242732168001353732062058237, 7.43878740065089690731465005624, 7.951382815875139330452511071140, 9.007941507733334403977455006928