L(s) = 1 | + (−0.923 − 0.382i)2-s + (0.707 + 0.707i)3-s + (0.707 + 0.707i)4-s + (−0.382 + 0.923i)5-s + (−0.382 − 0.923i)6-s + (−0.382 − 0.923i)8-s + 1.00i·9-s + (0.707 − 0.707i)10-s + 1.84·11-s + i·12-s + (0.707 − 0.707i)13-s + (−0.923 + 0.382i)15-s + i·16-s + (0.382 − 0.923i)18-s + (−0.923 + 0.382i)20-s + ⋯ |
L(s) = 1 | + (−0.923 − 0.382i)2-s + (0.707 + 0.707i)3-s + (0.707 + 0.707i)4-s + (−0.382 + 0.923i)5-s + (−0.382 − 0.923i)6-s + (−0.382 − 0.923i)8-s + 1.00i·9-s + (0.707 − 0.707i)10-s + 1.84·11-s + i·12-s + (0.707 − 0.707i)13-s + (−0.923 + 0.382i)15-s + i·16-s + (0.382 − 0.923i)18-s + (−0.923 + 0.382i)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
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9775525160\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9775525160\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.923 + 0.382i)T \) |
| 3 | \( 1 + (-0.707 - 0.707i)T \) |
| 5 | \( 1 + (0.382 - 0.923i)T \) |
| 13 | \( 1 + (-0.707 + 0.707i)T \) |
good | 7 | \( 1 - iT^{2} \) |
| 11 | \( 1 - 1.84T + T^{2} \) |
| 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 | \( 1 + 0.765T + T^{2} \) |
| 43 | \( 1 + (1 + i)T + iT^{2} \) |
| 47 | \( 1 + (-0.541 - 0.541i)T + iT^{2} \) |
| 53 | \( 1 + iT^{2} \) |
| 59 | \( 1 - 0.765iT - T^{2} \) |
| 61 | \( 1 + 1.41iT - T^{2} \) |
| 67 | \( 1 + iT^{2} \) |
| 71 | \( 1 - 1.84iT - T^{2} \) |
| 73 | \( 1 - iT^{2} \) |
| 79 | \( 1 + 1.41T + T^{2} \) |
| 83 | \( 1 + (1.30 + 1.30i)T + iT^{2} \) |
| 89 | \( 1 + 1.84iT - T^{2} \) |
| 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
−9.832477749268400040935804853365, −8.903824126043457525425866047617, −8.468720597271017211203288413375, −7.53083805384123347863071542421, −6.81937672568561567456697975403, −5.93267086526834280666277363491, −4.23684759866024398917128283392, −3.58320074206911322882441922879, −2.88916611630815101354547698073, −1.60209411274406067817478018437,
1.15159301427133153853303145341, 1.82634627406202565527851043338, 3.44225802900443672317375322903, 4.36955103686400381068940432251, 5.74987709920295435225094853615, 6.63887753353175902101982509450, 7.08769184002181774361200599214, 8.239492497640670037213606226068, 8.579551933220222464565108514569, 9.277268242423898897200997109598