L(s) = 1 | + (0.382 − 0.923i)2-s + (−0.707 − 0.707i)3-s + (−0.707 − 0.707i)4-s + (−0.923 − 0.382i)5-s + (−0.923 + 0.382i)6-s + (−0.923 + 0.382i)8-s + 1.00i·9-s + (−0.707 + 0.707i)10-s − 0.765·11-s + i·12-s + (−0.707 + 0.707i)13-s + (0.382 + 0.923i)15-s + i·16-s + (0.923 + 0.382i)18-s + (0.382 + 0.923i)20-s + ⋯ |
L(s) = 1 | + (0.382 − 0.923i)2-s + (−0.707 − 0.707i)3-s + (−0.707 − 0.707i)4-s + (−0.923 − 0.382i)5-s + (−0.923 + 0.382i)6-s + (−0.923 + 0.382i)8-s + 1.00i·9-s + (−0.707 + 0.707i)10-s − 0.765·11-s + i·12-s + (−0.707 + 0.707i)13-s + (0.382 + 0.923i)15-s + i·16-s + (0.923 + 0.382i)18-s + (0.382 + 0.923i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.03600595267\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.03600595267\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.382 + 0.923i)T \) |
| 3 | \( 1 + (0.707 + 0.707i)T \) |
| 5 | \( 1 + (0.923 + 0.382i)T \) |
| 13 | \( 1 + (0.707 - 0.707i)T \) |
good | 7 | \( 1 - iT^{2} \) |
| 11 | \( 1 + 0.765T + T^{2} \) |
| 17 | \( 1 + iT^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + iT^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 - iT^{2} \) |
| 41 | \( 1 + 1.84T + T^{2} \) |
| 43 | \( 1 + (1 + i)T + iT^{2} \) |
| 47 | \( 1 + (1.30 + 1.30i)T + iT^{2} \) |
| 53 | \( 1 + iT^{2} \) |
| 59 | \( 1 - 1.84iT - T^{2} \) |
| 61 | \( 1 - 1.41iT - T^{2} \) |
| 67 | \( 1 + iT^{2} \) |
| 71 | \( 1 + 0.765iT - T^{2} \) |
| 73 | \( 1 - iT^{2} \) |
| 79 | \( 1 - 1.41T + T^{2} \) |
| 83 | \( 1 + (0.541 + 0.541i)T + iT^{2} \) |
| 89 | \( 1 - 0.765iT - T^{2} \) |
| 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
−10.09937457137542853964339370872, −8.950516818072257689221149466374, −8.210241460526578833838617748031, −7.33053309586402376528658827347, −6.50074893205652887628610037362, −5.27821746417602055094332674090, −4.88750740784330323689534989067, −3.85871012120193172970736729992, −2.66010211980967207248313129505, −1.53247708731844805900247322929,
0.02752846837848939101940541531, 3.01605514531165697726192175769, 3.67899003066210673536977811417, 4.87542355465153552716459209181, 5.10103490551865509503419756936, 6.31553138430115086075046112170, 6.93042762632931199317700738666, 7.913182057958590935668531654454, 8.366580808764143767511821032633, 9.599448121608625535640581947664