L(s) = 1 | + (−2.66 − 1.10i)3-s + (5.50 + 13.2i)5-s + (6.48 − 6.48i)7-s + (−13.2 − 13.2i)9-s + (49.3 − 20.4i)11-s + (−21.4 + 51.7i)13-s − 41.4i·15-s − 3.73i·17-s + (−36.7 + 88.6i)19-s + (−24.4 + 10.1i)21-s + (45.4 + 45.4i)23-s + (−58.0 + 58.0i)25-s + (50.3 + 121. i)27-s + (−51.9 − 21.5i)29-s + 73.5·31-s + ⋯ |
L(s) = 1 | + (−0.512 − 0.212i)3-s + (0.492 + 1.18i)5-s + (0.349 − 0.349i)7-s + (−0.489 − 0.489i)9-s + (1.35 − 0.560i)11-s + (−0.457 + 1.10i)13-s − 0.714i·15-s − 0.0533i·17-s + (−0.443 + 1.07i)19-s + (−0.253 + 0.105i)21-s + (0.412 + 0.412i)23-s + (−0.464 + 0.464i)25-s + (0.359 + 0.867i)27-s + (−0.332 − 0.137i)29-s + 0.425·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.450 - 0.892i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.450 - 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.564226487\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.564226487\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
good | 3 | \( 1 + (2.66 + 1.10i)T + (19.0 + 19.0i)T^{2} \) |
| 5 | \( 1 + (-5.50 - 13.2i)T + (-88.3 + 88.3i)T^{2} \) |
| 7 | \( 1 + (-6.48 + 6.48i)T - 343iT^{2} \) |
| 11 | \( 1 + (-49.3 + 20.4i)T + (941. - 941. i)T^{2} \) |
| 13 | \( 1 + (21.4 - 51.7i)T + (-1.55e3 - 1.55e3i)T^{2} \) |
| 17 | \( 1 + 3.73iT - 4.91e3T^{2} \) |
| 19 | \( 1 + (36.7 - 88.6i)T + (-4.85e3 - 4.85e3i)T^{2} \) |
| 23 | \( 1 + (-45.4 - 45.4i)T + 1.21e4iT^{2} \) |
| 29 | \( 1 + (51.9 + 21.5i)T + (1.72e4 + 1.72e4i)T^{2} \) |
| 31 | \( 1 - 73.5T + 2.97e4T^{2} \) |
| 37 | \( 1 + (-165. - 399. i)T + (-3.58e4 + 3.58e4i)T^{2} \) |
| 41 | \( 1 + (-334. - 334. i)T + 6.89e4iT^{2} \) |
| 43 | \( 1 + (-328. + 136. i)T + (5.62e4 - 5.62e4i)T^{2} \) |
| 47 | \( 1 - 185. iT - 1.03e5T^{2} \) |
| 53 | \( 1 + (412. - 171. i)T + (1.05e5 - 1.05e5i)T^{2} \) |
| 59 | \( 1 + (214. + 518. i)T + (-1.45e5 + 1.45e5i)T^{2} \) |
| 61 | \( 1 + (85.1 + 35.2i)T + (1.60e5 + 1.60e5i)T^{2} \) |
| 67 | \( 1 + (252. + 104. i)T + (2.12e5 + 2.12e5i)T^{2} \) |
| 71 | \( 1 + (430. - 430. i)T - 3.57e5iT^{2} \) |
| 73 | \( 1 + (-41.8 - 41.8i)T + 3.89e5iT^{2} \) |
| 79 | \( 1 + 1.21e3iT - 4.93e5T^{2} \) |
| 83 | \( 1 + (290. - 702. i)T + (-4.04e5 - 4.04e5i)T^{2} \) |
| 89 | \( 1 + (-365. + 365. i)T - 7.04e5iT^{2} \) |
| 97 | \( 1 - 508.T + 9.12e5T^{2} \) |
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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
−11.47241430183522821544736328111, −11.10192821198974894443251317005, −9.863810056964360683035303591988, −9.017387754235406954770580048119, −7.58578397107116917701568533034, −6.38434376245727367840940994102, −6.18047246666817005215251120133, −4.35634377731581433836851812341, −3.04500808078470234133901191893, −1.38943824108587811059497194885,
0.72622701983175321790122340174, 2.31480220117366852941408732873, 4.37040663411854780882876996097, 5.19834169137403672685100095377, 6.04320969944836169435597688159, 7.50742997592744030937659877554, 8.778841819578170251700727193279, 9.278471822532456696371594178262, 10.53455742711176838247116180386, 11.43897298202369763599690152847