L(s) = 1 | − 2-s + (1.35 + 1.08i)3-s + 4-s + (−0.5 + 0.866i)5-s + (−1.35 − 1.08i)6-s + (2.32 − 1.26i)7-s − 8-s + (0.653 + 2.92i)9-s + (0.5 − 0.866i)10-s + (−0.975 − 1.68i)11-s + (1.35 + 1.08i)12-s + (2.18 + 3.77i)13-s + (−2.32 + 1.26i)14-s + (−1.61 + 0.629i)15-s + 16-s + (2.22 − 3.85i)17-s + ⋯ |
L(s) = 1 | − 0.707·2-s + (0.780 + 0.625i)3-s + 0.5·4-s + (−0.223 + 0.387i)5-s + (−0.551 − 0.442i)6-s + (0.878 − 0.478i)7-s − 0.353·8-s + (0.217 + 0.975i)9-s + (0.158 − 0.273i)10-s + (−0.294 − 0.509i)11-s + (0.390 + 0.312i)12-s + (0.605 + 1.04i)13-s + (−0.621 + 0.338i)14-s + (−0.416 + 0.162i)15-s + 0.250·16-s + (0.539 − 0.934i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.537 - 0.843i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.537 - 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.31724 + 0.722948i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.31724 + 0.722948i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + (-1.35 - 1.08i)T \) |
| 5 | \( 1 + (0.5 - 0.866i)T \) |
| 7 | \( 1 + (-2.32 + 1.26i)T \) |
good | 11 | \( 1 + (0.975 + 1.68i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.18 - 3.77i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-2.22 + 3.85i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.18 - 3.77i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.957 - 1.65i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.725 + 1.25i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 2.78T + 31T^{2} \) |
| 37 | \( 1 + (1.44 + 2.50i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-4.07 - 7.05i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (1.39 - 2.42i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 6.79T + 47T^{2} \) |
| 53 | \( 1 + (3.07 - 5.33i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + 8.10T + 59T^{2} \) |
| 61 | \( 1 - 0.523T + 61T^{2} \) |
| 67 | \( 1 + 11.0T + 67T^{2} \) |
| 71 | \( 1 - 16.3T + 71T^{2} \) |
| 73 | \( 1 + (-7.69 + 13.3i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 - 7.55T + 79T^{2} \) |
| 83 | \( 1 + (-1.48 + 2.58i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-4.62 - 8.01i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (6.19 - 10.7i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.67593615410207951073046366657, −9.753425576088030572012755866705, −9.081336250234452506562289260524, −7.904771313927550612447669050311, −7.79590820884555246943163064974, −6.46310174191804212110827317160, −5.10280951345298611155099028782, −3.98580701003944981584742847505, −2.96714645006397097404559514240, −1.55854703639707183198003228307,
1.10280953856654043222127451424, 2.28239644659945936398164086706, 3.49818007869526484475510014641, 4.99060973071115166459906690874, 6.11762027960766285675072163798, 7.28229580192691099071954782632, 8.114823553583234689798136388628, 8.433046772779006982255959324400, 9.372998917200258911859462379741, 10.34520962433208209868902261642