L(s) = 1 | − i·7-s + i·9-s + (1 − i)11-s − i·25-s + (1 + i)29-s + (1 − i)37-s + (−1 + i)43-s − 49-s + (−1 + i)53-s + 63-s + (−1 − i)67-s + 2i·71-s + (−1 − i)77-s − 81-s + (1 + i)99-s + ⋯ |
L(s) = 1 | − i·7-s + i·9-s + (1 − i)11-s − i·25-s + (1 + i)29-s + (1 − i)37-s + (−1 + i)43-s − 49-s + (−1 + i)53-s + 63-s + (−1 − i)67-s + 2i·71-s + (−1 − i)77-s − 81-s + (1 + i)99-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.042883170\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.042883170\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + iT \) |
good | 3 | \( 1 - iT^{2} \) |
| 5 | \( 1 + iT^{2} \) |
| 11 | \( 1 + (-1 + i)T - iT^{2} \) |
| 13 | \( 1 - iT^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - iT^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + (-1 - i)T + iT^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + (-1 + i)T - iT^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + (1 - i)T - iT^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + (1 - i)T - iT^{2} \) |
| 59 | \( 1 + iT^{2} \) |
| 61 | \( 1 - iT^{2} \) |
| 67 | \( 1 + (1 + i)T + iT^{2} \) |
| 71 | \( 1 - 2iT - T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 - iT^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−10.43096756719523208032377652804, −9.466837397912958160309878311214, −8.482925147518149956285616867902, −7.81229612672820194244209303087, −6.83070471734107728099826645424, −6.05282413147094061700808480869, −4.83454051071575640101649654839, −4.01707793218408609635329984241, −2.87648605440487255235932315351, −1.28183308675642035097784966116,
1.62432983907612950363498786813, 2.96291420653715139077888087295, 4.07256301127090740614578750652, 5.08929932523714480970487642022, 6.23265098550158758680943743417, 6.74456999986489418142346205143, 7.906721637694024845984422229152, 8.898482236773261366700062761677, 9.457780361990515816240673801009, 10.12646156634763419906207487499