| L(s) = 1 | + 1.73i·3-s + 2.55i·5-s + (6.95 − 0.748i)7-s − 2.99·9-s + 5.82·11-s + 21.8i·13-s − 4.42·15-s − 16.7i·17-s − 1.37i·19-s + (1.29 + 12.0i)21-s + 10.2·23-s + 18.4·25-s − 5.19i·27-s + 32.1·29-s + 35.1i·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + 0.511i·5-s + (0.994 − 0.106i)7-s − 0.333·9-s + 0.529·11-s + 1.67i·13-s − 0.295·15-s − 0.983i·17-s − 0.0722i·19-s + (0.0617 + 0.574i)21-s + 0.447·23-s + 0.738·25-s − 0.192i·27-s + 1.10·29-s + 1.13i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.106 - 0.994i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.106 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.227804169\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.227804169\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 1.73iT \) |
| 7 | \( 1 + (-6.95 + 0.748i)T \) |
| good | 5 | \( 1 - 2.55iT - 25T^{2} \) |
| 11 | \( 1 - 5.82T + 121T^{2} \) |
| 13 | \( 1 - 21.8iT - 169T^{2} \) |
| 17 | \( 1 + 16.7iT - 289T^{2} \) |
| 19 | \( 1 + 1.37iT - 361T^{2} \) |
| 23 | \( 1 - 10.2T + 529T^{2} \) |
| 29 | \( 1 - 32.1T + 841T^{2} \) |
| 31 | \( 1 - 35.1iT - 961T^{2} \) |
| 37 | \( 1 + 47.2T + 1.36e3T^{2} \) |
| 41 | \( 1 + 12.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 20.4T + 1.84e3T^{2} \) |
| 47 | \( 1 - 86.4iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 51.3T + 2.80e3T^{2} \) |
| 59 | \( 1 + 41.5iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 103. iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 25.5T + 4.48e3T^{2} \) |
| 71 | \( 1 - 74.8T + 5.04e3T^{2} \) |
| 73 | \( 1 + 6.02iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 20.8T + 6.24e3T^{2} \) |
| 83 | \( 1 - 141. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 137. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 56.0iT - 9.40e3T^{2} \) |
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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
−9.501047317750275120683440636241, −8.968002324215252887118287347292, −8.150653010580272008674917387100, −6.94654893014624009688112344440, −6.63863963170015865392140101298, −5.14412100041082639480528190151, −4.64621665500594043176242773661, −3.65583433838921408188143614257, −2.51736461018779686275167987092, −1.29264114720093189426675910369,
0.71095558984465277228253110661, 1.64188781652339768451396893475, 2.87946780786172906547443479555, 4.08339411074828436185207695537, 5.15485858470190454315687876143, 5.74496546480110529526638011402, 6.81808941805668289232326877121, 7.72055988752756837801307892875, 8.456595281566565520461586495159, 8.804919644975634442301034562591
