| 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
−11.12746938572771078369725617629, −10.32747661751738228765318238631, −9.101290262766874501303280777015, −7.85133960814485813131679618787, −6.60538219085896972429562469205, −5.99767890935893411005013684296, −5.30682220047606427759838424628, −4.32187935708822216577195303986, −2.65788409873924838370013541356, −2.15210761767179187942096281062,
2.16014952800002997846379505247, 3.57953569615054296965308662174, 4.03801608216190495666384666705, 5.37269422833808295805384855031, 6.10047065901310183920128631126, 6.71554413146774647192396636182, 7.989214244435098307099818247746, 9.496886277175972096218097959613, 10.60582064457982387607225706770, 10.91364520587742706013311151919
