L(s) = 1 | + (−0.415 − 0.909i)2-s + (−0.589 + 0.173i)3-s + (−0.654 + 0.755i)4-s + (3.36 + 2.16i)5-s + (0.402 + 0.464i)6-s + (−0.142 − 0.989i)7-s + (0.959 + 0.281i)8-s + (−2.20 + 1.41i)9-s + (0.569 − 3.95i)10-s + (−2.44 + 5.35i)11-s + (0.255 − 0.558i)12-s + (−0.0479 + 0.333i)13-s + (−0.841 + 0.540i)14-s + (−2.35 − 0.691i)15-s + (−0.142 − 0.989i)16-s + (1.96 + 2.27i)17-s + ⋯ |
L(s) = 1 | + (−0.293 − 0.643i)2-s + (−0.340 + 0.0999i)3-s + (−0.327 + 0.377i)4-s + (1.50 + 0.966i)5-s + (0.164 + 0.189i)6-s + (−0.0537 − 0.374i)7-s + (0.339 + 0.0996i)8-s + (−0.735 + 0.472i)9-s + (0.179 − 1.25i)10-s + (−0.736 + 1.61i)11-s + (0.0736 − 0.161i)12-s + (−0.0132 + 0.0924i)13-s + (−0.224 + 0.144i)14-s + (−0.608 − 0.178i)15-s + (−0.0355 − 0.247i)16-s + (0.477 + 0.551i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.815 - 0.578i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.815 - 0.578i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.06004 + 0.337478i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.06004 + 0.337478i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.415 + 0.909i)T \) |
| 7 | \( 1 + (0.142 + 0.989i)T \) |
| 23 | \( 1 + (-3.84 + 2.86i)T \) |
good | 3 | \( 1 + (0.589 - 0.173i)T + (2.52 - 1.62i)T^{2} \) |
| 5 | \( 1 + (-3.36 - 2.16i)T + (2.07 + 4.54i)T^{2} \) |
| 11 | \( 1 + (2.44 - 5.35i)T + (-7.20 - 8.31i)T^{2} \) |
| 13 | \( 1 + (0.0479 - 0.333i)T + (-12.4 - 3.66i)T^{2} \) |
| 17 | \( 1 + (-1.96 - 2.27i)T + (-2.41 + 16.8i)T^{2} \) |
| 19 | \( 1 + (-0.686 + 0.792i)T + (-2.70 - 18.8i)T^{2} \) |
| 29 | \( 1 + (0.265 + 0.306i)T + (-4.12 + 28.7i)T^{2} \) |
| 31 | \( 1 + (2.86 + 0.842i)T + (26.0 + 16.7i)T^{2} \) |
| 37 | \( 1 + (-8.42 + 5.41i)T + (15.3 - 33.6i)T^{2} \) |
| 41 | \( 1 + (-9.04 - 5.81i)T + (17.0 + 37.2i)T^{2} \) |
| 43 | \( 1 + (2.61 - 0.768i)T + (36.1 - 23.2i)T^{2} \) |
| 47 | \( 1 + 7.34T + 47T^{2} \) |
| 53 | \( 1 + (1.00 + 7.00i)T + (-50.8 + 14.9i)T^{2} \) |
| 59 | \( 1 + (-1.65 + 11.5i)T + (-56.6 - 16.6i)T^{2} \) |
| 61 | \( 1 + (5.30 + 1.55i)T + (51.3 + 32.9i)T^{2} \) |
| 67 | \( 1 + (-3.28 - 7.18i)T + (-43.8 + 50.6i)T^{2} \) |
| 71 | \( 1 + (0.274 + 0.601i)T + (-46.4 + 53.6i)T^{2} \) |
| 73 | \( 1 + (10.9 - 12.6i)T + (-10.3 - 72.2i)T^{2} \) |
| 79 | \( 1 + (-1.36 + 9.50i)T + (-75.7 - 22.2i)T^{2} \) |
| 83 | \( 1 + (-4.62 + 2.97i)T + (34.4 - 75.4i)T^{2} \) |
| 89 | \( 1 + (-6.07 + 1.78i)T + (74.8 - 48.1i)T^{2} \) |
| 97 | \( 1 + (2.44 + 1.57i)T + (40.2 + 88.2i)T^{2} \) |
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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
−11.34525432674816690462305376782, −10.68801070007354668348042570702, −9.993019528174763828024811775582, −9.403861803715435756711623964835, −7.900776350561975082530828412360, −6.87394839375628731502003918772, −5.76622552059708948203972116661, −4.71415933081512403364821386828, −2.88539862708241686746835170300, −1.98465639349360925864775579088,
0.959261644713516759794066414471, 2.90846200600161875694447966798, 5.12006183862043784727325399506, 5.69988961872761990416695383814, 6.25662418369106368435678438995, 7.87596848390307758206936832790, 8.923918148403552681374361427223, 9.291454514813420797346291418936, 10.45308987544120259347556444206, 11.47306808101525792212984681073