L(s) = 1 | − 3i·3-s − 5.86·5-s + (18.3 + 2.64i)7-s − 9·9-s + 34.5·11-s − 55.8·13-s + 17.5i·15-s + 2.77i·17-s + 67.8i·19-s + (7.94 − 54.9i)21-s − 176. i·23-s − 90.6·25-s + 27i·27-s + 116. i·29-s + 312.·31-s + ⋯ |
L(s) = 1 | − 0.577i·3-s − 0.524·5-s + (0.989 + 0.143i)7-s − 0.333·9-s + 0.946·11-s − 1.19·13-s + 0.302i·15-s + 0.0395i·17-s + 0.819i·19-s + (0.0826 − 0.571i)21-s − 1.59i·23-s − 0.724·25-s + 0.192i·27-s + 0.743i·29-s + 1.80·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.117 + 0.993i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.117 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.834629452\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.834629452\) |
\(L(\frac{5}{2})\) |
|
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 | \( 1 + (-18.3 - 2.64i)T \) |
good | 5 | \( 1 + 5.86T + 125T^{2} \) |
| 11 | \( 1 - 34.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 55.8T + 2.19e3T^{2} \) |
| 17 | \( 1 - 2.77iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 67.8iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 176. iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 116. iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 312.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 118. iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 280. iT - 6.89e4T^{2} \) |
| 43 | \( 1 - 15.1T + 7.95e4T^{2} \) |
| 47 | \( 1 - 6.34T + 1.03e5T^{2} \) |
| 53 | \( 1 + 23.2iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 288. iT - 2.05e5T^{2} \) |
| 61 | \( 1 - 514.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 295.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 475. iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 473. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 796. iT - 4.93e5T^{2} \) |
| 83 | \( 1 - 877. iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 33.8iT - 7.04e5T^{2} \) |
| 97 | \( 1 + 700. iT - 9.12e5T^{2} \) |
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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
−8.791962055682152712761920109337, −8.226155012463908978333595649436, −7.46400342568459291873377223909, −6.73781263510436757108890351236, −5.77018885326944076797230013392, −4.73110248145281459140923517565, −4.00975818181440843756612838717, −2.64521776037386951247826076897, −1.68858691174473775594161748404, −0.50709343072705565489100658958,
0.960505914525455393521701376273, 2.28110779337490108525266933572, 3.48752480374964706083028127949, 4.45452530050322177441632999551, 4.93112688340427429571714583008, 6.06273896801221161677933773158, 7.14915683313182072460416948968, 7.84308911046667281620460887601, 8.582830426145552111244589157177, 9.593386307528508353491340035673