L(s) = 1 | + 2·7-s + 2·13-s − 8·17-s − 6·19-s + 23-s − 5·25-s − 2·29-s + 4·31-s + 6·37-s − 10·41-s − 6·43-s − 3·49-s − 12·53-s + 4·59-s − 10·61-s + 6·67-s + 2·73-s + 6·79-s + 12·89-s + 4·91-s + 6·97-s − 6·101-s + 14·103-s − 14·109-s + 12·113-s − 16·119-s + ⋯ |
L(s) = 1 | + 0.755·7-s + 0.554·13-s − 1.94·17-s − 1.37·19-s + 0.208·23-s − 25-s − 0.371·29-s + 0.718·31-s + 0.986·37-s − 1.56·41-s − 0.914·43-s − 3/7·49-s − 1.64·53-s + 0.520·59-s − 1.28·61-s + 0.733·67-s + 0.234·73-s + 0.675·79-s + 1.27·89-s + 0.419·91-s + 0.609·97-s − 0.597·101-s + 1.37·103-s − 1.34·109-s + 1.12·113-s − 1.46·119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 - T \) |
good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 8 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 6 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−8.345569636812867160254308915571, −7.65594970385466816849743235118, −6.50179644485107772398564833778, −6.29774444113777872626330689118, −4.99792151603600177958797515416, −4.47853269014448701609330442713, −3.62304370073662129218815785945, −2.35528892230099071261582593666, −1.64013736344550873690026087923, 0,
1.64013736344550873690026087923, 2.35528892230099071261582593666, 3.62304370073662129218815785945, 4.47853269014448701609330442713, 4.99792151603600177958797515416, 6.29774444113777872626330689118, 6.50179644485107772398564833778, 7.65594970385466816849743235118, 8.345569636812867160254308915571