L(s) = 1 | − 3-s − 5-s + 7-s + 9-s − 3·11-s − 7·13-s + 15-s − 3·17-s + 19-s − 21-s + 3·23-s + 25-s − 27-s + 6·29-s + 10·31-s + 3·33-s − 35-s + 37-s + 7·39-s + 4·43-s − 45-s + 12·47-s − 6·49-s + 3·51-s + 9·53-s + 3·55-s − 57-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s − 0.904·11-s − 1.94·13-s + 0.258·15-s − 0.727·17-s + 0.229·19-s − 0.218·21-s + 0.625·23-s + 1/5·25-s − 0.192·27-s + 1.11·29-s + 1.79·31-s + 0.522·33-s − 0.169·35-s + 0.164·37-s + 1.12·39-s + 0.609·43-s − 0.149·45-s + 1.75·47-s − 6/7·49-s + 0.420·51-s + 1.23·53-s + 0.404·55-s − 0.132·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 37 | \( 1 - T \) |
good | 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 7 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−7.32629650959485491042657496775, −6.88412775586520352856561830185, −6.01029824046557045434889170922, −5.08044705706523656851697759248, −4.79785182526695662763552852044, −4.12540446297713192967190181758, −2.76753372863843207958498957752, −2.44704180441384093595733820451, −1.00924948765815190675846550013, 0,
1.00924948765815190675846550013, 2.44704180441384093595733820451, 2.76753372863843207958498957752, 4.12540446297713192967190181758, 4.79785182526695662763552852044, 5.08044705706523656851697759248, 6.01029824046557045434889170922, 6.88412775586520352856561830185, 7.32629650959485491042657496775