| L(s) = 1 | + 3·3-s − 2.14i·5-s + (17.7 + 5.16i)7-s + 9·9-s + 22.4i·11-s + 45.2i·13-s − 6.43i·15-s + 33.9i·17-s + 43.2·19-s + (53.3 + 15.4i)21-s − 42.1i·23-s + 120.·25-s + 27·27-s + 65.2·29-s − 188.·31-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.191i·5-s + (0.960 + 0.278i)7-s + 0.333·9-s + 0.615i·11-s + 0.966i·13-s − 0.110i·15-s + 0.484i·17-s + 0.522·19-s + (0.554 + 0.161i)21-s − 0.381i·23-s + 0.963·25-s + 0.192·27-s + 0.417·29-s − 1.09·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.278 - 0.960i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.278 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.850186464\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.850186464\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 3T \) |
| 7 | \( 1 + (-17.7 - 5.16i)T \) |
| good | 5 | \( 1 + 2.14iT - 125T^{2} \) |
| 11 | \( 1 - 22.4iT - 1.33e3T^{2} \) |
| 13 | \( 1 - 45.2iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 33.9iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 43.2T + 6.85e3T^{2} \) |
| 23 | \( 1 + 42.1iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 65.2T + 2.43e4T^{2} \) |
| 31 | \( 1 + 188.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 165.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 210. iT - 6.89e4T^{2} \) |
| 43 | \( 1 - 322. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 189.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 511.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 17.9T + 2.05e5T^{2} \) |
| 61 | \( 1 - 472. iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 565. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 595. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + 412. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 377. iT - 4.93e5T^{2} \) |
| 83 | \( 1 - 238.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 585. iT - 7.04e5T^{2} \) |
| 97 | \( 1 + 601. iT - 9.12e5T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.211855189988138564975956763541, −8.676139429195887862641880441343, −7.85105565286956958879501257618, −7.12027714627910299528571132678, −6.16616250701474111045120545679, −4.92626854046909273563439487728, −4.45917057893419006661275582994, −3.25270934709607075782982736178, −2.07532260758187420704162923643, −1.31899712926120335992356854841,
0.60660098222372414131857944531, 1.75326892069391634028305281826, 2.96012486703504303958658470557, 3.70361703089311486385681561486, 4.93481135595543406727024220173, 5.54613535603734411698240860848, 6.83938678389830795403100162826, 7.55879320013064784548067198874, 8.264954935677129685074617275087, 8.929056939745197262630807324699
