L(s) = 1 | + (0.316 − 5.18i)3-s + (5.49 + 9.52i)5-s + (6.85 + 17.2i)7-s + (−26.8 − 3.28i)9-s + (13.8 + 7.98i)11-s + (−77.4 + 44.7i)13-s + (51.1 − 25.4i)15-s − 106.·17-s + 8.31i·19-s + (91.3 − 30.1i)21-s + (−123. + 71.5i)23-s + (2.06 − 3.58i)25-s + (−25.4 + 137. i)27-s + (129. + 74.8i)29-s + (−37.1 + 21.4i)31-s + ⋯ |
L(s) = 1 | + (0.0608 − 0.998i)3-s + (0.491 + 0.851i)5-s + (0.370 + 0.928i)7-s + (−0.992 − 0.121i)9-s + (0.378 + 0.218i)11-s + (−1.65 + 0.953i)13-s + (0.879 − 0.438i)15-s − 1.52·17-s + 0.100i·19-s + (0.949 − 0.313i)21-s + (−1.12 + 0.648i)23-s + (0.0165 − 0.0286i)25-s + (−0.181 + 0.983i)27-s + (0.830 + 0.479i)29-s + (−0.215 + 0.124i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0828 - 0.996i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0828 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.200379703\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.200379703\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.316 + 5.18i)T \) |
| 7 | \( 1 + (-6.85 - 17.2i)T \) |
good | 5 | \( 1 + (-5.49 - 9.52i)T + (-62.5 + 108. i)T^{2} \) |
| 11 | \( 1 + (-13.8 - 7.98i)T + (665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + (77.4 - 44.7i)T + (1.09e3 - 1.90e3i)T^{2} \) |
| 17 | \( 1 + 106.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 8.31iT - 6.85e3T^{2} \) |
| 23 | \( 1 + (123. - 71.5i)T + (6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-129. - 74.8i)T + (1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 + (37.1 - 21.4i)T + (1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 - 390.T + 5.06e4T^{2} \) |
| 41 | \( 1 + (-172. - 298. i)T + (-3.44e4 + 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-28.5 + 49.4i)T + (-3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 + (-8.13 + 14.0i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + 445. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + (-193. - 334. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (420. + 242. i)T + (1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (251. + 436. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 - 751. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + 507. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + (-381. + 660. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + (607. - 1.05e3i)T + (-2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 - 425.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-494. - 285. i)T + (4.56e5 + 7.90e5i)T^{2} \) |
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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
−11.85901043935499541193273496135, −11.20954751912661904537494128353, −9.797906476999007682126651628518, −8.958369376049456538366478894681, −7.77161221012871008769629471219, −6.77017573675276954276842296015, −6.10660077421722982595539357439, −4.67856895763551585930080116266, −2.60905877754804987991339295534, −1.97998049698719125618679097500,
0.43693327890307940826251706425, 2.48068101119970447724486688557, 4.24569116645730382335895876747, 4.81926329469544574682354815275, 6.03282816659275282421930349267, 7.55779546507927516233589049940, 8.610363291714487101278565239948, 9.537397991127115973867380014456, 10.27724805474224288609600035596, 11.14653206823452050615777931330