L(s) = 1 | − 4.37·5-s − 7-s − 1.37·11-s + 2·13-s − 1.37·17-s − 5·19-s − 1.62·23-s + 14.1·25-s + 8.74·29-s − 2·31-s + 4.37·35-s + 2·37-s − 4.62·41-s + 8.11·43-s + 49-s + 8.74·53-s + 6·55-s − 10.1·59-s + 3.11·61-s − 8.74·65-s + 2.11·67-s − 7.11·71-s + 12.1·73-s + 1.37·77-s + 5.11·79-s + 17.4·83-s + 6·85-s + ⋯ |
L(s) = 1 | − 1.95·5-s − 0.377·7-s − 0.413·11-s + 0.554·13-s − 0.332·17-s − 1.14·19-s − 0.339·23-s + 2.82·25-s + 1.62·29-s − 0.359·31-s + 0.739·35-s + 0.328·37-s − 0.722·41-s + 1.23·43-s + 0.142·49-s + 1.20·53-s + 0.809·55-s − 1.31·59-s + 0.399·61-s − 1.08·65-s + 0.258·67-s − 0.844·71-s + 1.41·73-s + 0.156·77-s + 0.575·79-s + 1.91·83-s + 0.650·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9072 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 5 | \( 1 + 4.37T + 5T^{2} \) |
| 11 | \( 1 + 1.37T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + 1.37T + 17T^{2} \) |
| 19 | \( 1 + 5T + 19T^{2} \) |
| 23 | \( 1 + 1.62T + 23T^{2} \) |
| 29 | \( 1 - 8.74T + 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 + 4.62T + 41T^{2} \) |
| 43 | \( 1 - 8.11T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 8.74T + 53T^{2} \) |
| 59 | \( 1 + 10.1T + 59T^{2} \) |
| 61 | \( 1 - 3.11T + 61T^{2} \) |
| 67 | \( 1 - 2.11T + 67T^{2} \) |
| 71 | \( 1 + 7.11T + 71T^{2} \) |
| 73 | \( 1 - 12.1T + 73T^{2} \) |
| 79 | \( 1 - 5.11T + 79T^{2} \) |
| 83 | \( 1 - 17.4T + 83T^{2} \) |
| 89 | \( 1 + 14.7T + 89T^{2} \) |
| 97 | \( 1 + 8.11T + 97T^{2} \) |
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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
−7.49819129050945152678159805660, −6.74077671050710717338325794918, −6.24851272659363846945735161537, −5.14095031024772881740613222428, −4.38272672290212644294860081696, −3.93425098518615902304588292574, −3.19364928163553671669682340061, −2.38679141024468417545491990507, −0.912734859987117860545957080215, 0,
0.912734859987117860545957080215, 2.38679141024468417545491990507, 3.19364928163553671669682340061, 3.93425098518615902304588292574, 4.38272672290212644294860081696, 5.14095031024772881740613222428, 6.24851272659363846945735161537, 6.74077671050710717338325794918, 7.49819129050945152678159805660