L(s) = 1 | + 3-s + 4·7-s + 9-s + 6·13-s + 4·19-s + 4·21-s − 8·23-s + 27-s + 6·29-s − 6·37-s + 6·39-s − 10·41-s + 4·43-s − 8·47-s + 9·49-s − 10·53-s + 4·57-s − 6·61-s + 4·63-s + 4·67-s − 8·69-s − 14·73-s − 16·79-s + 81-s − 12·83-s + 6·87-s + 2·89-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 1.51·7-s + 1/3·9-s + 1.66·13-s + 0.917·19-s + 0.872·21-s − 1.66·23-s + 0.192·27-s + 1.11·29-s − 0.986·37-s + 0.960·39-s − 1.56·41-s + 0.609·43-s − 1.16·47-s + 9/7·49-s − 1.37·53-s + 0.529·57-s − 0.768·61-s + 0.503·63-s + 0.488·67-s − 0.963·69-s − 1.63·73-s − 1.80·79-s + 1/9·81-s − 1.31·83-s + 0.643·87-s + 0.211·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 173400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 173400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 17 | \( 1 \) |
good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−13.67919301525646, −13.05865452103205, −12.48626302387703, −11.80491567967824, −11.66217708704381, −11.11792129971497, −10.58852137367821, −10.13115452066023, −9.700684503218732, −8.905433121773147, −8.569805219091228, −8.165214290471001, −7.905398569589877, −7.238017215561196, −6.687965375566386, −6.037208306191244, −5.630923530312043, −4.946124203246173, −4.488565988801850, −3.972175844540059, −3.352705012730798, −2.903727263632836, −1.966125579028712, −1.512855970986462, −1.200112023937803, 0,
1.200112023937803, 1.512855970986462, 1.966125579028712, 2.903727263632836, 3.352705012730798, 3.972175844540059, 4.488565988801850, 4.946124203246173, 5.630923530312043, 6.037208306191244, 6.687965375566386, 7.238017215561196, 7.905398569589877, 8.165214290471001, 8.569805219091228, 8.905433121773147, 9.700684503218732, 10.13115452066023, 10.58852137367821, 11.11792129971497, 11.66217708704381, 11.80491567967824, 12.48626302387703, 13.05865452103205, 13.67919301525646