L(s) = 1 | + (2.56 + 0.103i)2-s + (−0.278 + 0.960i)3-s + (4.59 + 0.371i)4-s + (1.84 + 0.700i)5-s + (−0.814 + 2.43i)6-s + (−1.21 − 2.84i)7-s + (6.66 + 0.809i)8-s + (−0.845 − 0.534i)9-s + (4.67 + 1.99i)10-s + (−2.30 − 3.64i)11-s + (−1.63 + 4.31i)12-s + (1.98 + 3.01i)13-s + (−2.82 − 7.43i)14-s + (−1.18 + 1.58i)15-s + (7.94 + 1.29i)16-s + (−0.883 + 0.376i)17-s + ⋯ |
L(s) = 1 | + (1.81 + 0.0732i)2-s + (−0.160 + 0.554i)3-s + (2.29 + 0.185i)4-s + (0.826 + 0.313i)5-s + (−0.332 + 0.995i)6-s + (−0.458 − 1.07i)7-s + (2.35 + 0.286i)8-s + (−0.281 − 0.178i)9-s + (1.47 + 0.629i)10-s + (−0.695 − 1.10i)11-s + (−0.472 + 1.24i)12-s + (0.550 + 0.834i)13-s + (−0.754 − 1.98i)14-s + (−0.306 + 0.407i)15-s + (1.98 + 0.322i)16-s + (−0.214 + 0.0913i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.914 - 0.403i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.914 - 0.403i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.87693 + 0.817600i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.87693 + 0.817600i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.278 - 0.960i)T \) |
| 13 | \( 1 + (-1.98 - 3.01i)T \) |
good | 2 | \( 1 + (-2.56 - 0.103i)T + (1.99 + 0.160i)T^{2} \) |
| 5 | \( 1 + (-1.84 - 0.700i)T + (3.74 + 3.31i)T^{2} \) |
| 7 | \( 1 + (1.21 + 2.84i)T + (-4.84 + 5.04i)T^{2} \) |
| 11 | \( 1 + (2.30 + 3.64i)T + (-4.71 + 9.93i)T^{2} \) |
| 17 | \( 1 + (0.883 - 0.376i)T + (11.7 - 12.2i)T^{2} \) |
| 19 | \( 1 + (5.15 - 2.97i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (3.68 - 6.37i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (0.109 - 2.71i)T + (-28.9 - 2.33i)T^{2} \) |
| 31 | \( 1 + (1.82 + 2.05i)T + (-3.73 + 30.7i)T^{2} \) |
| 37 | \( 1 + (-0.170 + 0.0347i)T + (34.0 - 14.5i)T^{2} \) |
| 41 | \( 1 + (-8.35 - 2.41i)T + (34.6 + 21.9i)T^{2} \) |
| 43 | \( 1 + (-1.09 + 5.37i)T + (-39.5 - 16.8i)T^{2} \) |
| 47 | \( 1 + (-0.0305 + 0.0210i)T + (16.6 - 43.9i)T^{2} \) |
| 53 | \( 1 + (-1.35 + 11.1i)T + (-51.4 - 12.6i)T^{2} \) |
| 59 | \( 1 + (-0.0265 - 0.163i)T + (-55.9 + 18.6i)T^{2} \) |
| 61 | \( 1 + (0.00409 - 0.00307i)T + (16.9 - 58.5i)T^{2} \) |
| 67 | \( 1 + (-0.885 - 10.9i)T + (-66.1 + 10.7i)T^{2} \) |
| 71 | \( 1 + (-9.67 + 9.29i)T + (2.85 - 70.9i)T^{2} \) |
| 73 | \( 1 + (-7.47 + 14.2i)T + (-41.4 - 60.0i)T^{2} \) |
| 79 | \( 1 + (-5.90 - 8.55i)T + (-28.0 + 73.8i)T^{2} \) |
| 83 | \( 1 + (1.04 - 4.25i)T + (-73.4 - 38.5i)T^{2} \) |
| 89 | \( 1 + (-11.1 - 6.41i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-3.66 - 2.99i)T + (19.4 + 95.0i)T^{2} \) |
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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
−10.91364520587742706013311151919, −10.60582064457982387607225706770, −9.496886277175972096218097959613, −7.989214244435098307099818247746, −6.71554413146774647192396636182, −6.10047065901310183920128631126, −5.37269422833808295805384855031, −4.03801608216190495666384666705, −3.57953569615054296965308662174, −2.16014952800002997846379505247,
2.15210761767179187942096281062, 2.65788409873924838370013541356, 4.32187935708822216577195303986, 5.30682220047606427759838424628, 5.99767890935893411005013684296, 6.60538219085896972429562469205, 7.85133960814485813131679618787, 9.101290262766874501303280777015, 10.32747661751738228765318238631, 11.12746938572771078369725617629