L(s) = 1 | − 3-s + 5-s + 2·7-s + 9-s − 4·11-s − 15-s + 17-s + 4·19-s − 2·21-s + 25-s − 27-s + 8·31-s + 4·33-s + 2·35-s + 2·37-s − 2·41-s − 6·43-s + 45-s − 12·47-s − 3·49-s − 51-s + 12·53-s − 4·55-s − 4·57-s + 2·63-s + 12·67-s − 12·71-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.755·7-s + 1/3·9-s − 1.20·11-s − 0.258·15-s + 0.242·17-s + 0.917·19-s − 0.436·21-s + 1/5·25-s − 0.192·27-s + 1.43·31-s + 0.696·33-s + 0.338·35-s + 0.328·37-s − 0.312·41-s − 0.914·43-s + 0.149·45-s − 1.75·47-s − 3/7·49-s − 0.140·51-s + 1.64·53-s − 0.539·55-s − 0.529·57-s + 0.251·63-s + 1.46·67-s − 1.42·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 344760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 344760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 13 | \( 1 \) |
| 17 | \( 1 - T \) |
good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 12 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 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
−12.94568744632769, −12.17078094063677, −11.78053514320365, −11.56800758185494, −10.94942956832072, −10.52353611722286, −10.03831704528576, −9.843055543360987, −9.245292913382313, −8.467791378326921, −8.247968568731865, −7.751482387467634, −7.259077756224003, −6.745663272493530, −6.222123264187902, −5.662069607994103, −5.307958204954907, −4.741563612096554, −4.634890655332637, −3.673586846695884, −3.141478327237891, −2.577590639405045, −1.991981345572964, −1.361512506569495, −0.8136921604492044, 0,
0.8136921604492044, 1.361512506569495, 1.991981345572964, 2.577590639405045, 3.141478327237891, 3.673586846695884, 4.634890655332637, 4.741563612096554, 5.307958204954907, 5.662069607994103, 6.222123264187902, 6.745663272493530, 7.259077756224003, 7.751482387467634, 8.247968568731865, 8.467791378326921, 9.245292913382313, 9.843055543360987, 10.03831704528576, 10.52353611722286, 10.94942956832072, 11.56800758185494, 11.78053514320365, 12.17078094063677, 12.94568744632769