| L(s) = 1 | + (1.36 + 0.358i)2-s + (−1.11 + 0.779i)3-s + (1.74 + 0.980i)4-s + (−3.17 + 0.277i)5-s + (−1.80 + 0.667i)6-s + (−0.306 + 0.176i)7-s + (2.03 + 1.96i)8-s + (−0.394 + 1.08i)9-s + (−4.43 − 0.757i)10-s + (1.06 + 3.99i)11-s + (−2.70 + 0.266i)12-s + (−2.57 + 3.68i)13-s + (−0.482 + 0.132i)14-s + (3.31 − 2.77i)15-s + (2.07 + 3.41i)16-s + (5.28 − 1.92i)17-s + ⋯ |
| L(s) = 1 | + (0.967 + 0.253i)2-s + (−0.642 + 0.449i)3-s + (0.871 + 0.490i)4-s + (−1.41 + 0.124i)5-s + (−0.735 + 0.272i)6-s + (−0.115 + 0.0668i)7-s + (0.718 + 0.695i)8-s + (−0.131 + 0.361i)9-s + (−1.40 − 0.239i)10-s + (0.322 + 1.20i)11-s + (−0.780 + 0.0769i)12-s + (−0.714 + 1.02i)13-s + (−0.128 + 0.0353i)14-s + (0.855 − 0.717i)15-s + (0.518 + 0.854i)16-s + (1.28 − 0.466i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.498 - 0.866i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.498 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.678018 + 1.17261i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.678018 + 1.17261i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.36 - 0.358i)T \) |
| 19 | \( 1 + (0.299 + 4.34i)T \) |
| good | 3 | \( 1 + (1.11 - 0.779i)T + (1.02 - 2.81i)T^{2} \) |
| 5 | \( 1 + (3.17 - 0.277i)T + (4.92 - 0.868i)T^{2} \) |
| 7 | \( 1 + (0.306 - 0.176i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.06 - 3.99i)T + (-9.52 + 5.5i)T^{2} \) |
| 13 | \( 1 + (2.57 - 3.68i)T + (-4.44 - 12.2i)T^{2} \) |
| 17 | \( 1 + (-5.28 + 1.92i)T + (13.0 - 10.9i)T^{2} \) |
| 23 | \( 1 + (3.70 + 4.41i)T + (-3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (-3.03 - 6.51i)T + (-18.6 + 22.2i)T^{2} \) |
| 31 | \( 1 + (1.77 + 3.07i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-4.96 + 4.96i)T - 37iT^{2} \) |
| 41 | \( 1 + (-8.60 - 1.51i)T + (38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (-7.15 + 0.626i)T + (42.3 - 7.46i)T^{2} \) |
| 47 | \( 1 + (-8.85 - 3.22i)T + (36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + (0.935 - 10.6i)T + (-52.1 - 9.20i)T^{2} \) |
| 59 | \( 1 + (2.22 - 4.77i)T + (-37.9 - 45.1i)T^{2} \) |
| 61 | \( 1 + (8.82 + 0.772i)T + (60.0 + 10.5i)T^{2} \) |
| 67 | \( 1 + (2.27 + 4.87i)T + (-43.0 + 51.3i)T^{2} \) |
| 71 | \( 1 + (3.64 - 4.34i)T + (-12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + (-0.0104 - 0.00184i)T + (68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (-1.87 + 10.6i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (-1.15 + 4.30i)T + (-71.8 - 41.5i)T^{2} \) |
| 89 | \( 1 + (8.54 - 1.50i)T + (83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (-3.97 + 1.44i)T + (74.3 - 62.3i)T^{2} \) |
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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
−12.27514456407454646933499510091, −11.31663788419686861022510087853, −10.60374962211660088538725858280, −9.256807566750616507789756432233, −7.63487849811027235733470572314, −7.29946363561364851882916961003, −5.96275323628887027405134647879, −4.51514236547225020612644318723, −4.37955237642509047899939726565, −2.66457839278947286008489113405,
0.813734211580066864382233190277, 3.24551191414810671060936668097, 3.96128436489932715627883235861, 5.52815124513544053431668946842, 6.14070589911227544738249558553, 7.50874333903473237062186624584, 8.103408705968858949026329737139, 9.880384533849901210932553784649, 10.92740365475665084153507590854, 11.78734481803836172206010458964
