L(s) = 1 | + (−0.707 − 0.707i)2-s + (1.30 + 1.30i)3-s + 1.00i·4-s + (0.923 + 0.382i)5-s − 1.84i·6-s + (0.707 − 0.707i)8-s + 2.41i·9-s + (−0.382 − 0.923i)10-s + (−1.30 + 1.30i)12-s + (−0.541 − 0.541i)13-s + (0.707 + 1.70i)15-s − 1.00·16-s + (1.70 − 1.70i)18-s − 0.765·19-s + (−0.382 + 0.923i)20-s + ⋯ |
L(s) = 1 | + (−0.707 − 0.707i)2-s + (1.30 + 1.30i)3-s + 1.00i·4-s + (0.923 + 0.382i)5-s − 1.84i·6-s + (0.707 − 0.707i)8-s + 2.41i·9-s + (−0.382 − 0.923i)10-s + (−1.30 + 1.30i)12-s + (−0.541 − 0.541i)13-s + (0.707 + 1.70i)15-s − 1.00·16-s + (1.70 − 1.70i)18-s − 0.765·19-s + (−0.382 + 0.923i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.584 - 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.584 - 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.428224617\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.428224617\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 + 0.707i)T \) |
| 5 | \( 1 + (-0.923 - 0.382i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (-1.30 - 1.30i)T + iT^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 + (0.541 + 0.541i)T + iT^{2} \) |
| 17 | \( 1 + iT^{2} \) |
| 19 | \( 1 + 0.765T + T^{2} \) |
| 23 | \( 1 + (-1 + i)T - iT^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 - iT^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 + iT^{2} \) |
| 53 | \( 1 + iT^{2} \) |
| 59 | \( 1 + 1.84T + T^{2} \) |
| 61 | \( 1 + 1.84iT - T^{2} \) |
| 67 | \( 1 - iT^{2} \) |
| 71 | \( 1 + 1.41T + T^{2} \) |
| 73 | \( 1 - iT^{2} \) |
| 79 | \( 1 + 1.41iT - T^{2} \) |
| 83 | \( 1 + (-0.541 - 0.541i)T + iT^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + iT^{2} \) |
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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
−9.339307722413369425934876298506, −9.099419973380599849809065007881, −8.262262278158031216621242948399, −7.55373263640289557081333430545, −6.49321663201466847667464683112, −5.08372625184832739370626747581, −4.38185669204567361948563505248, −3.27361357961696287485301442741, −2.73822029929784028844690006331, −1.91549915719754936815807246213,
1.26036351910011343751498745850, 1.99139846824909120045518665821, 2.89929859717625044337624414899, 4.45984546038362720506879316248, 5.62752648938212831169448802090, 6.41393128536192617747070708613, 7.08599176562412332943108379978, 7.65795768745808642153017316493, 8.555612032339964593949407240147, 9.030474709024362152585500078749