| L(s) = 1 | + (−1.66 − 0.174i)2-s + (−1.65 − 0.500i)3-s + (0.771 + 0.163i)4-s + (1.94 − 1.10i)5-s + (2.66 + 1.12i)6-s + (−0.907 + 0.523i)7-s + (1.92 + 0.625i)8-s + (2.49 + 1.66i)9-s + (−3.42 + 1.49i)10-s + (0.524 − 4.98i)11-s + (−1.19 − 0.657i)12-s + (−5.17 + 0.543i)13-s + (1.59 − 0.711i)14-s + (−3.77 + 0.856i)15-s + (−4.52 − 2.01i)16-s + (−2.45 − 0.798i)17-s + ⋯ |
| L(s) = 1 | + (−1.17 − 0.123i)2-s + (−0.957 − 0.289i)3-s + (0.385 + 0.0819i)4-s + (0.869 − 0.493i)5-s + (1.08 + 0.457i)6-s + (−0.342 + 0.197i)7-s + (0.680 + 0.221i)8-s + (0.832 + 0.553i)9-s + (−1.08 + 0.472i)10-s + (0.158 − 1.50i)11-s + (−0.345 − 0.189i)12-s + (−1.43 + 0.150i)13-s + (0.427 − 0.190i)14-s + (−0.975 + 0.221i)15-s + (−1.13 − 0.503i)16-s + (−0.596 − 0.193i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.825 + 0.564i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.825 + 0.564i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0914736 - 0.296035i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0914736 - 0.296035i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.65 + 0.500i)T \) |
| 5 | \( 1 + (-1.94 + 1.10i)T \) |
| good | 2 | \( 1 + (1.66 + 0.174i)T + (1.95 + 0.415i)T^{2} \) |
| 7 | \( 1 + (0.907 - 0.523i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.524 + 4.98i)T + (-10.7 - 2.28i)T^{2} \) |
| 13 | \( 1 + (5.17 - 0.543i)T + (12.7 - 2.70i)T^{2} \) |
| 17 | \( 1 + (2.45 + 0.798i)T + (13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (0.420 - 1.29i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (2.88 + 6.48i)T + (-15.3 + 17.0i)T^{2} \) |
| 29 | \( 1 + (2.86 - 3.18i)T + (-3.03 - 28.8i)T^{2} \) |
| 31 | \( 1 + (4.45 + 4.94i)T + (-3.24 + 30.8i)T^{2} \) |
| 37 | \( 1 + (-1.74 - 2.40i)T + (-11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (-0.295 - 2.80i)T + (-40.1 + 8.52i)T^{2} \) |
| 43 | \( 1 + (-0.434 + 0.250i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (4.17 + 3.76i)T + (4.91 + 46.7i)T^{2} \) |
| 53 | \( 1 + (-13.1 + 4.26i)T + (42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (-0.192 - 1.83i)T + (-57.7 + 12.2i)T^{2} \) |
| 61 | \( 1 + (-1.13 + 10.7i)T + (-59.6 - 12.6i)T^{2} \) |
| 67 | \( 1 + (-0.899 + 0.810i)T + (7.00 - 66.6i)T^{2} \) |
| 71 | \( 1 + (3.62 + 11.1i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-0.855 + 1.17i)T + (-22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (2.69 - 2.98i)T + (-8.25 - 78.5i)T^{2} \) |
| 83 | \( 1 + (-0.159 - 0.751i)T + (-75.8 + 33.7i)T^{2} \) |
| 89 | \( 1 + (13.8 + 10.0i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-11.0 - 9.93i)T + (10.1 + 96.4i)T^{2} \) |
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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
−11.62895721524347866604278166708, −10.68808651861515449425458428335, −9.887228732554397114504770191635, −9.080265686637628627736702640656, −8.078428152973241637676181333462, −6.78859527085951323049984527434, −5.76857910685672875393946338773, −4.68317960759708409027850022391, −2.11633488129868413970013530690, −0.41072938411884692892481043854,
1.92356890356657319845940828488, 4.31070147335546335465694067096, 5.51272200313389578681075583710, 6.99566177630040469075483100255, 7.29804447822907667483583226651, 9.193096281391135780152615906624, 9.883939388669206011200790719622, 10.22082766698625935005819284636, 11.35451916193452297414136846584, 12.53245340746900870178380830603
