L(s) = 1 | − 3-s − 5-s + 9-s − 6·11-s + 15-s + 6·17-s + 4·19-s + 6·23-s + 25-s − 27-s − 2·29-s − 8·31-s + 6·33-s − 2·37-s + 10·41-s − 12·43-s − 45-s + 8·47-s − 6·51-s − 2·53-s + 6·55-s − 4·57-s + 4·59-s − 8·61-s − 16·67-s − 6·69-s − 10·71-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1/3·9-s − 1.80·11-s + 0.258·15-s + 1.45·17-s + 0.917·19-s + 1.25·23-s + 1/5·25-s − 0.192·27-s − 0.371·29-s − 1.43·31-s + 1.04·33-s − 0.328·37-s + 1.56·41-s − 1.82·43-s − 0.149·45-s + 1.16·47-s − 0.840·51-s − 0.274·53-s + 0.809·55-s − 0.529·57-s + 0.520·59-s − 1.02·61-s − 1.95·67-s − 0.722·69-s − 1.18·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−8.195997459266505541820217552576, −7.46376856014807308263488802456, −7.18423018991062681617685488028, −5.75453195504572146888016273486, −5.41905713103952371212453657091, −4.65177918460320059407287261226, −3.46169360080650512626267915520, −2.77036736691272232935634878251, −1.30149973513856413707143112892, 0,
1.30149973513856413707143112892, 2.77036736691272232935634878251, 3.46169360080650512626267915520, 4.65177918460320059407287261226, 5.41905713103952371212453657091, 5.75453195504572146888016273486, 7.18423018991062681617685488028, 7.46376856014807308263488802456, 8.195997459266505541820217552576